Properties

Label 40-92e20-1.1-c1e20-0-0
Degree $40$
Conductor $1.887\times 10^{39}$
Sign $1$
Analytic cond. $0.00209555$
Root an. cond. $0.857101$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 2·5-s + 2·7-s + 3·9-s − 2·11-s + 6·13-s + 4·15-s − 9·17-s − 11·19-s + 4·21-s − 22·23-s − 25-s − 5·27-s − 29-s − 13·31-s − 4·33-s + 4·35-s + 34·37-s + 12·39-s + 28·41-s + 44·43-s + 6·45-s + 26·47-s + 39·49-s − 18·51-s + 14·53-s − 4·55-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s + 0.755·7-s + 9-s − 0.603·11-s + 1.66·13-s + 1.03·15-s − 2.18·17-s − 2.52·19-s + 0.872·21-s − 4.58·23-s − 1/5·25-s − 0.962·27-s − 0.185·29-s − 2.33·31-s − 0.696·33-s + 0.676·35-s + 5.58·37-s + 1.92·39-s + 4.37·41-s + 6.70·43-s + 0.894·45-s + 3.79·47-s + 39/7·49-s − 2.52·51-s + 1.92·53-s − 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 23^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 23^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(2^{40} \cdot 23^{20}\)
Sign: $1$
Analytic conductor: \(0.00209555\)
Root analytic conductor: \(0.857101\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{92} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 2^{40} \cdot 23^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.8776286557\)
\(L(\frac12)\) \(\approx\) \(0.8776286557\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 + 22 T + 243 T^{2} + 1683 T^{3} + 7184 T^{4} + 12716 T^{5} - 61786 T^{6} - 645898 T^{7} - 3299537 T^{8} - 13775025 T^{9} - 61105010 T^{10} - 13775025 p T^{11} - 3299537 p^{2} T^{12} - 645898 p^{3} T^{13} - 61786 p^{4} T^{14} + 12716 p^{5} T^{15} + 7184 p^{6} T^{16} + 1683 p^{7} T^{17} + 243 p^{8} T^{18} + 22 p^{9} T^{19} + p^{10} T^{20} \)
good3 \( 1 - 2 T + T^{2} + p^{2} T^{3} - 2 p^{2} T^{4} - 23 T^{5} + 103 T^{6} - 44 T^{7} - 350 T^{8} + 817 T^{9} + 359 T^{10} - 88 p^{3} T^{11} + 1681 T^{12} + 6271 T^{13} - 4042 p T^{14} - 4823 T^{15} + 32407 T^{16} - 18892 T^{17} - 24457 T^{18} + 28585 T^{19} + 109678 T^{20} + 28585 p T^{21} - 24457 p^{2} T^{22} - 18892 p^{3} T^{23} + 32407 p^{4} T^{24} - 4823 p^{5} T^{25} - 4042 p^{7} T^{26} + 6271 p^{7} T^{27} + 1681 p^{8} T^{28} - 88 p^{12} T^{29} + 359 p^{10} T^{30} + 817 p^{11} T^{31} - 350 p^{12} T^{32} - 44 p^{13} T^{33} + 103 p^{14} T^{34} - 23 p^{15} T^{35} - 2 p^{18} T^{36} + p^{19} T^{37} + p^{18} T^{38} - 2 p^{19} T^{39} + p^{20} T^{40} \)
5 \( 1 - 2 T + p T^{2} - 33 T^{3} + 44 T^{4} - 187 T^{5} + 447 T^{6} - 498 T^{7} + 3104 T^{8} - 1749 T^{9} + 1397 p T^{10} - 37596 T^{11} - 117 p^{2} T^{12} - 154561 T^{13} + 285398 T^{14} - 32681 T^{15} + 2358763 T^{16} + 99588 p T^{17} + 3646397 T^{18} - 19523141 T^{19} - 35022054 T^{20} - 19523141 p T^{21} + 3646397 p^{2} T^{22} + 99588 p^{4} T^{23} + 2358763 p^{4} T^{24} - 32681 p^{5} T^{25} + 285398 p^{6} T^{26} - 154561 p^{7} T^{27} - 117 p^{10} T^{28} - 37596 p^{9} T^{29} + 1397 p^{11} T^{30} - 1749 p^{11} T^{31} + 3104 p^{12} T^{32} - 498 p^{13} T^{33} + 447 p^{14} T^{34} - 187 p^{15} T^{35} + 44 p^{16} T^{36} - 33 p^{17} T^{37} + p^{19} T^{38} - 2 p^{19} T^{39} + p^{20} T^{40} \)
7 \( 1 - 2 T - 5 p T^{2} + 68 T^{3} + 669 T^{4} - 1312 T^{5} - 9427 T^{6} + 18670 T^{7} + 109397 T^{8} - 214628 T^{9} - 1095970 T^{10} + 2081644 T^{11} + 1393653 p T^{12} - 16996895 T^{13} - 11323491 p T^{14} + 113731908 T^{15} + 85896940 p T^{16} - 577629022 T^{17} - 4361760915 T^{18} + 1481292787 T^{19} + 30848041228 T^{20} + 1481292787 p T^{21} - 4361760915 p^{2} T^{22} - 577629022 p^{3} T^{23} + 85896940 p^{5} T^{24} + 113731908 p^{5} T^{25} - 11323491 p^{7} T^{26} - 16996895 p^{7} T^{27} + 1393653 p^{9} T^{28} + 2081644 p^{9} T^{29} - 1095970 p^{10} T^{30} - 214628 p^{11} T^{31} + 109397 p^{12} T^{32} + 18670 p^{13} T^{33} - 9427 p^{14} T^{34} - 1312 p^{15} T^{35} + 669 p^{16} T^{36} + 68 p^{17} T^{37} - 5 p^{19} T^{38} - 2 p^{19} T^{39} + p^{20} T^{40} \)
11 \( 1 + 2 T - 18 T^{2} - 91 T^{3} + 71 T^{4} + 1594 T^{5} + 1373 T^{6} - 9728 T^{7} - 36550 T^{8} + 56491 T^{9} + 508663 T^{10} + 20039 p T^{11} - 534744 p T^{12} - 2476251 p T^{13} + 4733042 p T^{14} + 33011354 p T^{15} + 3204471 p^{2} T^{16} - 20749649 p^{2} T^{17} - 64566368 p^{2} T^{18} + 96067529 p^{2} T^{19} + 385665534 p^{2} T^{20} + 96067529 p^{3} T^{21} - 64566368 p^{4} T^{22} - 20749649 p^{5} T^{23} + 3204471 p^{6} T^{24} + 33011354 p^{6} T^{25} + 4733042 p^{7} T^{26} - 2476251 p^{8} T^{27} - 534744 p^{9} T^{28} + 20039 p^{10} T^{29} + 508663 p^{10} T^{30} + 56491 p^{11} T^{31} - 36550 p^{12} T^{32} - 9728 p^{13} T^{33} + 1373 p^{14} T^{34} + 1594 p^{15} T^{35} + 71 p^{16} T^{36} - 91 p^{17} T^{37} - 18 p^{18} T^{38} + 2 p^{19} T^{39} + p^{20} T^{40} \)
13 \( 1 - 6 T + 5 T^{2} + 6 p T^{3} - 665 T^{4} + 174 p T^{5} + 939 T^{6} - 3244 p T^{7} + 178847 T^{8} - 136680 T^{9} - 2003708 T^{10} + 9332716 T^{11} - 7153451 T^{12} - 104412947 T^{13} + 554304921 T^{14} - 507603394 T^{15} - 6982617428 T^{16} + 35030167370 T^{17} - 45011204005 T^{18} - 294988226711 T^{19} + 1854526873536 T^{20} - 294988226711 p T^{21} - 45011204005 p^{2} T^{22} + 35030167370 p^{3} T^{23} - 6982617428 p^{4} T^{24} - 507603394 p^{5} T^{25} + 554304921 p^{6} T^{26} - 104412947 p^{7} T^{27} - 7153451 p^{8} T^{28} + 9332716 p^{9} T^{29} - 2003708 p^{10} T^{30} - 136680 p^{11} T^{31} + 178847 p^{12} T^{32} - 3244 p^{14} T^{33} + 939 p^{14} T^{34} + 174 p^{16} T^{35} - 665 p^{16} T^{36} + 6 p^{18} T^{37} + 5 p^{18} T^{38} - 6 p^{19} T^{39} + p^{20} T^{40} \)
17 \( 1 + 9 T + 49 T^{2} - 24 T^{3} - 1213 T^{4} - 6859 T^{5} - 1291 T^{6} + 127215 T^{7} + 651189 T^{8} + 546557 T^{9} - 395459 p T^{10} - 28336781 T^{11} - 22279938 T^{12} - 20073716 T^{13} - 1425030724 T^{14} - 4464991763 T^{15} + 33596470298 T^{16} + 339449971028 T^{17} + 654071160281 T^{18} - 3936926670754 T^{19} - 33857770130546 T^{20} - 3936926670754 p T^{21} + 654071160281 p^{2} T^{22} + 339449971028 p^{3} T^{23} + 33596470298 p^{4} T^{24} - 4464991763 p^{5} T^{25} - 1425030724 p^{6} T^{26} - 20073716 p^{7} T^{27} - 22279938 p^{8} T^{28} - 28336781 p^{9} T^{29} - 395459 p^{11} T^{30} + 546557 p^{11} T^{31} + 651189 p^{12} T^{32} + 127215 p^{13} T^{33} - 1291 p^{14} T^{34} - 6859 p^{15} T^{35} - 1213 p^{16} T^{36} - 24 p^{17} T^{37} + 49 p^{18} T^{38} + 9 p^{19} T^{39} + p^{20} T^{40} \)
19 \( 1 + 11 T + 87 T^{2} + 451 T^{3} + 1873 T^{4} + 11 p^{2} T^{5} - 19563 T^{6} - 302511 T^{7} - 2124320 T^{8} - 10650068 T^{9} - 39426694 T^{10} - 79030897 T^{11} + 262406253 T^{12} + 3986944951 T^{13} + 25296171453 T^{14} + 112812218233 T^{15} + 342131008577 T^{16} + 353757923409 T^{17} - 4200738637139 T^{18} - 38378843129506 T^{19} - 200979811012320 T^{20} - 38378843129506 p T^{21} - 4200738637139 p^{2} T^{22} + 353757923409 p^{3} T^{23} + 342131008577 p^{4} T^{24} + 112812218233 p^{5} T^{25} + 25296171453 p^{6} T^{26} + 3986944951 p^{7} T^{27} + 262406253 p^{8} T^{28} - 79030897 p^{9} T^{29} - 39426694 p^{10} T^{30} - 10650068 p^{11} T^{31} - 2124320 p^{12} T^{32} - 302511 p^{13} T^{33} - 19563 p^{14} T^{34} + 11 p^{17} T^{35} + 1873 p^{16} T^{36} + 451 p^{17} T^{37} + 87 p^{18} T^{38} + 11 p^{19} T^{39} + p^{20} T^{40} \)
29 \( 1 + T - 89 T^{2} + 389 T^{3} + 3869 T^{4} - 1405 p T^{5} - 9675 T^{6} + 1922227 T^{7} - 6912898 T^{8} - 40982276 T^{9} + 398579942 T^{10} - 149408099 T^{11} - 10934479101 T^{12} + 39217894769 T^{13} + 135138673219 T^{14} - 1176063252329 T^{15} + 1161938151221 T^{16} + 18546180454435 T^{17} - 54822434672349 T^{18} - 109233431939124 T^{19} + 1249030195547208 T^{20} - 109233431939124 p T^{21} - 54822434672349 p^{2} T^{22} + 18546180454435 p^{3} T^{23} + 1161938151221 p^{4} T^{24} - 1176063252329 p^{5} T^{25} + 135138673219 p^{6} T^{26} + 39217894769 p^{7} T^{27} - 10934479101 p^{8} T^{28} - 149408099 p^{9} T^{29} + 398579942 p^{10} T^{30} - 40982276 p^{11} T^{31} - 6912898 p^{12} T^{32} + 1922227 p^{13} T^{33} - 9675 p^{14} T^{34} - 1405 p^{16} T^{35} + 3869 p^{16} T^{36} + 389 p^{17} T^{37} - 89 p^{18} T^{38} + p^{19} T^{39} + p^{20} T^{40} \)
31 \( 1 + 13 T + 67 T^{2} + 927 T^{3} + 9627 T^{4} + 48779 T^{5} + 411685 T^{6} + 3714993 T^{7} + 19441048 T^{8} + 124388876 T^{9} + 32559268 p T^{10} + 5548833813 T^{11} + 29902473527 T^{12} + 226468458711 T^{13} + 1325440592241 T^{14} + 6557231997811 T^{15} + 45797465482857 T^{16} + 282772631070121 T^{17} + 1361320537937067 T^{18} + 8251063919291516 T^{19} + 52615790958441368 T^{20} + 8251063919291516 p T^{21} + 1361320537937067 p^{2} T^{22} + 282772631070121 p^{3} T^{23} + 45797465482857 p^{4} T^{24} + 6557231997811 p^{5} T^{25} + 1325440592241 p^{6} T^{26} + 226468458711 p^{7} T^{27} + 29902473527 p^{8} T^{28} + 5548833813 p^{9} T^{29} + 32559268 p^{11} T^{30} + 124388876 p^{11} T^{31} + 19441048 p^{12} T^{32} + 3714993 p^{13} T^{33} + 411685 p^{14} T^{34} + 48779 p^{15} T^{35} + 9627 p^{16} T^{36} + 927 p^{17} T^{37} + 67 p^{18} T^{38} + 13 p^{19} T^{39} + p^{20} T^{40} \)
37 \( 1 - 34 T + 450 T^{2} - 2399 T^{3} - 1835 T^{4} + 26508 T^{5} + 1099105 T^{6} - 14869850 T^{7} + 67128414 T^{8} + 1747065 p T^{9} - 1025364467 T^{10} - 14860774893 T^{11} + 215273261312 T^{12} - 935984035999 T^{13} - 1036408961006 T^{14} + 19256252465244 T^{15} + 92188048558317 T^{16} - 1962812365479443 T^{17} + 10140091240136334 T^{18} - 240613611115303 T^{19} - 212734557705212530 T^{20} - 240613611115303 p T^{21} + 10140091240136334 p^{2} T^{22} - 1962812365479443 p^{3} T^{23} + 92188048558317 p^{4} T^{24} + 19256252465244 p^{5} T^{25} - 1036408961006 p^{6} T^{26} - 935984035999 p^{7} T^{27} + 215273261312 p^{8} T^{28} - 14860774893 p^{9} T^{29} - 1025364467 p^{10} T^{30} + 1747065 p^{12} T^{31} + 67128414 p^{12} T^{32} - 14869850 p^{13} T^{33} + 1099105 p^{14} T^{34} + 26508 p^{15} T^{35} - 1835 p^{16} T^{36} - 2399 p^{17} T^{37} + 450 p^{18} T^{38} - 34 p^{19} T^{39} + p^{20} T^{40} \)
41 \( 1 - 28 T + 234 T^{2} + 1232 T^{3} - 39629 T^{4} + 266163 T^{5} + 891360 T^{6} - 29035741 T^{7} + 200368785 T^{8} + 205072195 T^{9} - 15017046640 T^{10} + 115264447019 T^{11} - 58740585672 T^{12} - 6026283053463 T^{13} + 50587910156489 T^{14} - 84283098774316 T^{15} - 1841488783987446 T^{16} + 17934196357517076 T^{17} - 48586738660984363 T^{18} - 443795551398635537 T^{19} + 5234410776096907938 T^{20} - 443795551398635537 p T^{21} - 48586738660984363 p^{2} T^{22} + 17934196357517076 p^{3} T^{23} - 1841488783987446 p^{4} T^{24} - 84283098774316 p^{5} T^{25} + 50587910156489 p^{6} T^{26} - 6026283053463 p^{7} T^{27} - 58740585672 p^{8} T^{28} + 115264447019 p^{9} T^{29} - 15017046640 p^{10} T^{30} + 205072195 p^{11} T^{31} + 200368785 p^{12} T^{32} - 29035741 p^{13} T^{33} + 891360 p^{14} T^{34} + 266163 p^{15} T^{35} - 39629 p^{16} T^{36} + 1232 p^{17} T^{37} + 234 p^{18} T^{38} - 28 p^{19} T^{39} + p^{20} T^{40} \)
43 \( 1 - 44 T + 968 T^{2} - 14256 T^{3} + 159433 T^{4} - 1444894 T^{5} + 10957474 T^{6} - 71372037 T^{7} + 422248960 T^{8} - 2529200553 T^{9} + 16848885812 T^{10} - 118373511927 T^{11} + 736412913950 T^{12} - 3053172194391 T^{13} - 2975319060923 T^{14} + 209208704212354 T^{15} - 2449458285394295 T^{16} + 19732890802816576 T^{17} - 129708700440028347 T^{18} + 779966942959205320 T^{19} - 4861318231104484898 T^{20} + 779966942959205320 p T^{21} - 129708700440028347 p^{2} T^{22} + 19732890802816576 p^{3} T^{23} - 2449458285394295 p^{4} T^{24} + 209208704212354 p^{5} T^{25} - 2975319060923 p^{6} T^{26} - 3053172194391 p^{7} T^{27} + 736412913950 p^{8} T^{28} - 118373511927 p^{9} T^{29} + 16848885812 p^{10} T^{30} - 2529200553 p^{11} T^{31} + 422248960 p^{12} T^{32} - 71372037 p^{13} T^{33} + 10957474 p^{14} T^{34} - 1444894 p^{15} T^{35} + 159433 p^{16} T^{36} - 14256 p^{17} T^{37} + 968 p^{18} T^{38} - 44 p^{19} T^{39} + p^{20} T^{40} \)
47 \( ( 1 - 13 T + 331 T^{2} - 3934 T^{3} + 55143 T^{4} - 577380 T^{5} + 5897564 T^{6} - 53798482 T^{7} + 442777672 T^{8} - 3497125775 T^{9} + 24240466834 T^{10} - 3497125775 p T^{11} + 442777672 p^{2} T^{12} - 53798482 p^{3} T^{13} + 5897564 p^{4} T^{14} - 577380 p^{5} T^{15} + 55143 p^{6} T^{16} - 3934 p^{7} T^{17} + 331 p^{8} T^{18} - 13 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
53 \( 1 - 14 T - 54 T^{2} + 808 T^{3} + 10361 T^{4} - 36132 T^{5} - 618846 T^{6} - 6374361 T^{7} + 58684372 T^{8} + 450371347 T^{9} + 647023776 T^{10} - 37108711689 T^{11} - 139723521950 T^{12} + 198169407399 T^{13} + 17852406699347 T^{14} + 25911875234368 T^{15} - 170292975695561 T^{16} - 6602127807017918 T^{17} - 5484894184047039 T^{18} + 59633859111998184 T^{19} + 2336937123881917762 T^{20} + 59633859111998184 p T^{21} - 5484894184047039 p^{2} T^{22} - 6602127807017918 p^{3} T^{23} - 170292975695561 p^{4} T^{24} + 25911875234368 p^{5} T^{25} + 17852406699347 p^{6} T^{26} + 198169407399 p^{7} T^{27} - 139723521950 p^{8} T^{28} - 37108711689 p^{9} T^{29} + 647023776 p^{10} T^{30} + 450371347 p^{11} T^{31} + 58684372 p^{12} T^{32} - 6374361 p^{13} T^{33} - 618846 p^{14} T^{34} - 36132 p^{15} T^{35} + 10361 p^{16} T^{36} + 808 p^{17} T^{37} - 54 p^{18} T^{38} - 14 p^{19} T^{39} + p^{20} T^{40} \)
59 \( 1 + 10 T - 73 T^{2} - 770 T^{3} + 10274 T^{4} + 62843 T^{5} - 470897 T^{6} + 856337 T^{7} + 45006915 T^{8} - 199704714 T^{9} - 619120205 T^{10} + 36894460637 T^{11} + 83688495124 T^{12} - 2000233429722 T^{13} + 7232510348002 T^{14} + 161120283130136 T^{15} - 349710644448990 T^{16} - 4695796182069177 T^{17} + 69670216591859901 T^{18} + 225833951036228964 T^{19} - 2829443064508935832 T^{20} + 225833951036228964 p T^{21} + 69670216591859901 p^{2} T^{22} - 4695796182069177 p^{3} T^{23} - 349710644448990 p^{4} T^{24} + 161120283130136 p^{5} T^{25} + 7232510348002 p^{6} T^{26} - 2000233429722 p^{7} T^{27} + 83688495124 p^{8} T^{28} + 36894460637 p^{9} T^{29} - 619120205 p^{10} T^{30} - 199704714 p^{11} T^{31} + 45006915 p^{12} T^{32} + 856337 p^{13} T^{33} - 470897 p^{14} T^{34} + 62843 p^{15} T^{35} + 10274 p^{16} T^{36} - 770 p^{17} T^{37} - 73 p^{18} T^{38} + 10 p^{19} T^{39} + p^{20} T^{40} \)
61 \( 1 + 56 T + 1392 T^{2} + 21148 T^{3} + 237881 T^{4} + 2241022 T^{5} + 17295274 T^{6} + 97986115 T^{7} + 402306826 T^{8} + 1673178023 T^{9} + 12718509936 T^{10} + 202337940485 T^{11} + 3416725875458 T^{12} + 41643875255565 T^{13} + 384372429007179 T^{14} + 3101386209679348 T^{15} + 23169203491112305 T^{16} + 152853109978608688 T^{17} + 927044544608202715 T^{18} + 6276708491278371190 T^{19} + 48404660263252487842 T^{20} + 6276708491278371190 p T^{21} + 927044544608202715 p^{2} T^{22} + 152853109978608688 p^{3} T^{23} + 23169203491112305 p^{4} T^{24} + 3101386209679348 p^{5} T^{25} + 384372429007179 p^{6} T^{26} + 41643875255565 p^{7} T^{27} + 3416725875458 p^{8} T^{28} + 202337940485 p^{9} T^{29} + 12718509936 p^{10} T^{30} + 1673178023 p^{11} T^{31} + 402306826 p^{12} T^{32} + 97986115 p^{13} T^{33} + 17295274 p^{14} T^{34} + 2241022 p^{15} T^{35} + 237881 p^{16} T^{36} + 21148 p^{17} T^{37} + 1392 p^{18} T^{38} + 56 p^{19} T^{39} + p^{20} T^{40} \)
67 \( 1 + 44 T + 876 T^{2} + 10670 T^{3} + 82218 T^{4} + 184558 T^{5} - 5138255 T^{6} - 84481419 T^{7} - 672959892 T^{8} - 1681894456 T^{9} + 34805149677 T^{10} + 588920148256 T^{11} + 5114254464942 T^{12} + 25303007358299 T^{13} - 15213178207520 T^{14} - 1607253457130728 T^{15} - 16333743345868059 T^{16} - 79902082701476673 T^{17} + 223252489442772702 T^{18} + 8418012694471073265 T^{19} + 89206408148967963052 T^{20} + 8418012694471073265 p T^{21} + 223252489442772702 p^{2} T^{22} - 79902082701476673 p^{3} T^{23} - 16333743345868059 p^{4} T^{24} - 1607253457130728 p^{5} T^{25} - 15213178207520 p^{6} T^{26} + 25303007358299 p^{7} T^{27} + 5114254464942 p^{8} T^{28} + 588920148256 p^{9} T^{29} + 34805149677 p^{10} T^{30} - 1681894456 p^{11} T^{31} - 672959892 p^{12} T^{32} - 84481419 p^{13} T^{33} - 5138255 p^{14} T^{34} + 184558 p^{15} T^{35} + 82218 p^{16} T^{36} + 10670 p^{17} T^{37} + 876 p^{18} T^{38} + 44 p^{19} T^{39} + p^{20} T^{40} \)
71 \( 1 + 37 T + 753 T^{2} + 10281 T^{3} + 111601 T^{4} + 1027391 T^{5} + 7844739 T^{6} + 37877327 T^{7} - 81242002 T^{8} - 4615820560 T^{9} - 72952867522 T^{10} - 887463645749 T^{11} - 9046322732955 T^{12} - 75317082008605 T^{13} - 503129693862237 T^{14} - 2350657411261331 T^{15} + 792225855483947 T^{16} + 203287596699828997 T^{17} + 3176650429666579275 T^{18} + 34267150783868025244 T^{19} + \)\(30\!\cdots\!00\)\( T^{20} + 34267150783868025244 p T^{21} + 3176650429666579275 p^{2} T^{22} + 203287596699828997 p^{3} T^{23} + 792225855483947 p^{4} T^{24} - 2350657411261331 p^{5} T^{25} - 503129693862237 p^{6} T^{26} - 75317082008605 p^{7} T^{27} - 9046322732955 p^{8} T^{28} - 887463645749 p^{9} T^{29} - 72952867522 p^{10} T^{30} - 4615820560 p^{11} T^{31} - 81242002 p^{12} T^{32} + 37877327 p^{13} T^{33} + 7844739 p^{14} T^{34} + 1027391 p^{15} T^{35} + 111601 p^{16} T^{36} + 10281 p^{17} T^{37} + 753 p^{18} T^{38} + 37 p^{19} T^{39} + p^{20} T^{40} \)
73 \( 1 + 12 T - 111 T^{2} - 2042 T^{3} - 4678 T^{4} + 98867 T^{5} + 2088245 T^{6} + 12954545 T^{7} - 151701927 T^{8} - 2299090546 T^{9} - 3574200683 T^{10} + 1933143161 p T^{11} + 1614974076720 T^{12} + 2097156447186 T^{13} - 131826351778660 T^{14} - 1071918170202108 T^{15} + 2418273616457854 T^{16} + 92687936016053143 T^{17} + 598761279975652065 T^{18} - 2904943610669862562 T^{19} - 69553880347760229908 T^{20} - 2904943610669862562 p T^{21} + 598761279975652065 p^{2} T^{22} + 92687936016053143 p^{3} T^{23} + 2418273616457854 p^{4} T^{24} - 1071918170202108 p^{5} T^{25} - 131826351778660 p^{6} T^{26} + 2097156447186 p^{7} T^{27} + 1614974076720 p^{8} T^{28} + 1933143161 p^{10} T^{29} - 3574200683 p^{10} T^{30} - 2299090546 p^{11} T^{31} - 151701927 p^{12} T^{32} + 12954545 p^{13} T^{33} + 2088245 p^{14} T^{34} + 98867 p^{15} T^{35} - 4678 p^{16} T^{36} - 2042 p^{17} T^{37} - 111 p^{18} T^{38} + 12 p^{19} T^{39} + p^{20} T^{40} \)
79 \( 1 + 6 T - 3 p T^{2} - 3488 T^{3} - 5164 T^{4} + 466118 T^{5} + 7211868 T^{6} + 26928548 T^{7} - 587298208 T^{8} - 9988116276 T^{9} - 58001195134 T^{10} + 295734014832 T^{11} + 9344939905040 T^{12} + 82381410425084 T^{13} + 147235367371108 T^{14} - 4905671789061416 T^{15} - 71322076831251417 T^{16} - 473365610047334774 T^{17} - 11432648536864597 T^{18} + 35657100786519606066 T^{19} + \)\(43\!\cdots\!12\)\( T^{20} + 35657100786519606066 p T^{21} - 11432648536864597 p^{2} T^{22} - 473365610047334774 p^{3} T^{23} - 71322076831251417 p^{4} T^{24} - 4905671789061416 p^{5} T^{25} + 147235367371108 p^{6} T^{26} + 82381410425084 p^{7} T^{27} + 9344939905040 p^{8} T^{28} + 295734014832 p^{9} T^{29} - 58001195134 p^{10} T^{30} - 9988116276 p^{11} T^{31} - 587298208 p^{12} T^{32} + 26928548 p^{13} T^{33} + 7211868 p^{14} T^{34} + 466118 p^{15} T^{35} - 5164 p^{16} T^{36} - 3488 p^{17} T^{37} - 3 p^{19} T^{38} + 6 p^{19} T^{39} + p^{20} T^{40} \)
83 \( 1 + 25 T + 521 T^{2} + 11235 T^{3} + 171456 T^{4} + 2629040 T^{5} + 37078622 T^{6} + 444433600 T^{7} + 5493313895 T^{8} + 58886274897 T^{9} + 586628162678 T^{10} + 5796349201463 T^{11} + 45938576620517 T^{12} + 357956139666011 T^{13} + 2145092286352423 T^{14} + 3341981287539608 T^{15} - 70948383511782686 T^{16} - 2127337953493682732 T^{17} - 28046828336224116796 T^{18} - \)\(29\!\cdots\!85\)\( T^{19} - \)\(31\!\cdots\!22\)\( T^{20} - \)\(29\!\cdots\!85\)\( p T^{21} - 28046828336224116796 p^{2} T^{22} - 2127337953493682732 p^{3} T^{23} - 70948383511782686 p^{4} T^{24} + 3341981287539608 p^{5} T^{25} + 2145092286352423 p^{6} T^{26} + 357956139666011 p^{7} T^{27} + 45938576620517 p^{8} T^{28} + 5796349201463 p^{9} T^{29} + 586628162678 p^{10} T^{30} + 58886274897 p^{11} T^{31} + 5493313895 p^{12} T^{32} + 444433600 p^{13} T^{33} + 37078622 p^{14} T^{34} + 2629040 p^{15} T^{35} + 171456 p^{16} T^{36} + 11235 p^{17} T^{37} + 521 p^{18} T^{38} + 25 p^{19} T^{39} + p^{20} T^{40} \)
89 \( 1 - 10 T - 210 T^{2} + 867 T^{3} + 45903 T^{4} - 76448 T^{5} - 4487317 T^{6} - 14591644 T^{7} + 303798758 T^{8} + 1660241231 T^{9} + 14574338691 T^{10} - 70445308957 T^{11} - 4082685108656 T^{12} - 29229351823483 T^{13} + 479444343170728 T^{14} + 4903953764069050 T^{15} - 19633841659862311 T^{16} - 493652686249372825 T^{17} - 433127124395593868 T^{18} + 16308271365442618595 T^{19} + \)\(18\!\cdots\!78\)\( T^{20} + 16308271365442618595 p T^{21} - 433127124395593868 p^{2} T^{22} - 493652686249372825 p^{3} T^{23} - 19633841659862311 p^{4} T^{24} + 4903953764069050 p^{5} T^{25} + 479444343170728 p^{6} T^{26} - 29229351823483 p^{7} T^{27} - 4082685108656 p^{8} T^{28} - 70445308957 p^{9} T^{29} + 14574338691 p^{10} T^{30} + 1660241231 p^{11} T^{31} + 303798758 p^{12} T^{32} - 14591644 p^{13} T^{33} - 4487317 p^{14} T^{34} - 76448 p^{15} T^{35} + 45903 p^{16} T^{36} + 867 p^{17} T^{37} - 210 p^{18} T^{38} - 10 p^{19} T^{39} + p^{20} T^{40} \)
97 \( 1 + T + 27 T^{2} - 1170 T^{3} - 5501 T^{4} - 98921 T^{5} - 57159 T^{6} + 7456373 T^{7} + 12493481 T^{8} + 1124410641 T^{9} - 5511160665 T^{10} + 117167882957 T^{11} - 242089566110 T^{12} + 6747573764216 T^{13} - 96053925332208 T^{14} - 647782802765057 T^{15} - 5439550260847510 T^{16} - 87215804143099200 T^{17} + 887865892387155549 T^{18} - 1206831047047458096 T^{19} + \)\(14\!\cdots\!06\)\( T^{20} - 1206831047047458096 p T^{21} + 887865892387155549 p^{2} T^{22} - 87215804143099200 p^{3} T^{23} - 5439550260847510 p^{4} T^{24} - 647782802765057 p^{5} T^{25} - 96053925332208 p^{6} T^{26} + 6747573764216 p^{7} T^{27} - 242089566110 p^{8} T^{28} + 117167882957 p^{9} T^{29} - 5511160665 p^{10} T^{30} + 1124410641 p^{11} T^{31} + 12493481 p^{12} T^{32} + 7456373 p^{13} T^{33} - 57159 p^{14} T^{34} - 98921 p^{15} T^{35} - 5501 p^{16} T^{36} - 1170 p^{17} T^{37} + 27 p^{18} T^{38} + p^{19} T^{39} + p^{20} T^{40} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.07707823344088101802071971990, −3.94254543242120956861776676779, −3.80512979761728211225752170457, −3.79525083884723156320009140956, −3.60377068539049339525285314324, −3.57041507171523526268934603683, −3.48230188305601206118291930612, −3.09436552775246597526899220966, −3.08828830180894641792282261638, −2.91650208968170250628165228900, −2.87271157771041116109145278792, −2.75816976085574177475349359227, −2.59150141901499813509101343603, −2.53009620560665999772594151532, −2.44406715768035351040481065609, −2.42830682292154710101942505897, −2.39888837798450508560481424104, −2.35401475583982768211390930939, −2.09251117208412343917962359886, −1.99822007706356907414247232382, −1.88651142605817688110252411733, −1.50865022472740446597481635932, −1.38451886126928944133462712852, −1.28018652023636663535204959689, −1.22216756229914468491495340579, 1.22216756229914468491495340579, 1.28018652023636663535204959689, 1.38451886126928944133462712852, 1.50865022472740446597481635932, 1.88651142605817688110252411733, 1.99822007706356907414247232382, 2.09251117208412343917962359886, 2.35401475583982768211390930939, 2.39888837798450508560481424104, 2.42830682292154710101942505897, 2.44406715768035351040481065609, 2.53009620560665999772594151532, 2.59150141901499813509101343603, 2.75816976085574177475349359227, 2.87271157771041116109145278792, 2.91650208968170250628165228900, 3.08828830180894641792282261638, 3.09436552775246597526899220966, 3.48230188305601206118291930612, 3.57041507171523526268934603683, 3.60377068539049339525285314324, 3.79525083884723156320009140956, 3.80512979761728211225752170457, 3.94254543242120956861776676779, 4.07707823344088101802071971990

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.