Properties

Label 32-930e16-1.1-c1e16-0-9
Degree $32$
Conductor $3.131\times 10^{47}$
Sign $1$
Analytic cond. $8.55383\times 10^{13}$
Root an. cond. $2.72508$
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  − 4·2-s + 2·3-s + 6·4-s + 8·5-s − 8·6-s + 11·7-s − 4·8-s + 3·9-s − 32·10-s − 4·11-s + 12·12-s − 20·13-s − 44·14-s + 16·15-s + 16-s − 12·17-s − 12·18-s − 7·19-s + 48·20-s + 22·21-s + 16·22-s + 11·23-s − 8·24-s + 28·25-s + 80·26-s + 2·27-s + 66·28-s + ⋯
L(s)  = 1  − 2.82·2-s + 1.15·3-s + 3·4-s + 3.57·5-s − 3.26·6-s + 4.15·7-s − 1.41·8-s + 9-s − 10.1·10-s − 1.20·11-s + 3.46·12-s − 5.54·13-s − 11.7·14-s + 4.13·15-s + 1/4·16-s − 2.91·17-s − 2.82·18-s − 1.60·19-s + 10.7·20-s + 4.80·21-s + 3.41·22-s + 2.29·23-s − 1.63·24-s + 28/5·25-s + 15.6·26-s + 0.384·27-s + 12.4·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 31^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 31^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 31^{16}\)
Sign: $1$
Analytic conductor: \(8.55383\times 10^{13}\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 31^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(22.46123960\)
\(L(\frac12)\) \(\approx\) \(22.46123960\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
3 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
5 \( ( 1 - T + T^{2} )^{8} \)
31 \( 1 + 17 T + 111 T^{2} + 108 T^{3} - 2639 T^{4} - 15134 T^{5} - 26406 T^{6} - 71541 T^{7} - 735033 T^{8} - 71541 p T^{9} - 26406 p^{2} T^{10} - 15134 p^{3} T^{11} - 2639 p^{4} T^{12} + 108 p^{5} T^{13} + 111 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16} \)
good7 \( 1 - 11 T + 68 T^{2} - 264 T^{3} + 646 T^{4} - 645 T^{5} - 1685 T^{6} + 8318 T^{7} - 10008 T^{8} - 41414 T^{9} + 309006 T^{10} - 177650 p T^{11} + 543161 p T^{12} - 8273131 T^{13} + 1152933 p T^{14} + 21842249 T^{15} - 108641161 T^{16} + 21842249 p T^{17} + 1152933 p^{3} T^{18} - 8273131 p^{3} T^{19} + 543161 p^{5} T^{20} - 177650 p^{6} T^{21} + 309006 p^{6} T^{22} - 41414 p^{7} T^{23} - 10008 p^{8} T^{24} + 8318 p^{9} T^{25} - 1685 p^{10} T^{26} - 645 p^{11} T^{27} + 646 p^{12} T^{28} - 264 p^{13} T^{29} + 68 p^{14} T^{30} - 11 p^{15} T^{31} + p^{16} T^{32} \)
11 \( 1 + 4 T + 6 T^{2} - 25 T^{3} - 30 p T^{4} - 1568 T^{5} - 4007 T^{6} - 6328 T^{7} + 12850 T^{8} + 212265 T^{9} + 1048566 T^{10} + 353684 p T^{11} + 10094906 T^{12} + 6974945 T^{13} - 77687285 T^{14} - 518548747 T^{15} - 2103630623 T^{16} - 518548747 p T^{17} - 77687285 p^{2} T^{18} + 6974945 p^{3} T^{19} + 10094906 p^{4} T^{20} + 353684 p^{6} T^{21} + 1048566 p^{6} T^{22} + 212265 p^{7} T^{23} + 12850 p^{8} T^{24} - 6328 p^{9} T^{25} - 4007 p^{10} T^{26} - 1568 p^{11} T^{27} - 30 p^{13} T^{28} - 25 p^{13} T^{29} + 6 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \)
13 \( 1 + 20 T + 216 T^{2} + 1635 T^{3} + 9684 T^{4} + 47910 T^{5} + 209221 T^{6} + 859395 T^{7} + 3547196 T^{8} + 15247360 T^{9} + 67301658 T^{10} + 292447310 T^{11} + 1207006546 T^{12} + 4655162165 T^{13} + 1301090426 p T^{14} + 4589272355 p T^{15} + 212238154397 T^{16} + 4589272355 p^{2} T^{17} + 1301090426 p^{3} T^{18} + 4655162165 p^{3} T^{19} + 1207006546 p^{4} T^{20} + 292447310 p^{5} T^{21} + 67301658 p^{6} T^{22} + 15247360 p^{7} T^{23} + 3547196 p^{8} T^{24} + 859395 p^{9} T^{25} + 209221 p^{10} T^{26} + 47910 p^{11} T^{27} + 9684 p^{12} T^{28} + 1635 p^{13} T^{29} + 216 p^{14} T^{30} + 20 p^{15} T^{31} + p^{16} T^{32} \)
17 \( 1 + 12 T + 70 T^{2} + 94 T^{3} - 83 p T^{4} - 11342 T^{5} - 34265 T^{6} + 64608 T^{7} + 1154372 T^{8} + 5770758 T^{9} + 10227780 T^{10} - 56703710 T^{11} - 559548309 T^{12} - 2204391320 T^{13} - 1962167061 T^{14} + 29284605774 T^{15} + 192965144075 T^{16} + 29284605774 p T^{17} - 1962167061 p^{2} T^{18} - 2204391320 p^{3} T^{19} - 559548309 p^{4} T^{20} - 56703710 p^{5} T^{21} + 10227780 p^{6} T^{22} + 5770758 p^{7} T^{23} + 1154372 p^{8} T^{24} + 64608 p^{9} T^{25} - 34265 p^{10} T^{26} - 11342 p^{11} T^{27} - 83 p^{13} T^{28} + 94 p^{13} T^{29} + 70 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \)
19 \( 1 + 7 T + 54 T^{2} + 265 T^{3} + 1645 T^{4} + 9121 T^{5} + 44737 T^{6} + 234144 T^{7} + 1031295 T^{8} + 5232840 T^{9} + 21818044 T^{10} + 96530833 T^{11} + 399784861 T^{12} + 91279650 p T^{13} + 7815791245 T^{14} + 29724773786 T^{15} + 132009794377 T^{16} + 29724773786 p T^{17} + 7815791245 p^{2} T^{18} + 91279650 p^{4} T^{19} + 399784861 p^{4} T^{20} + 96530833 p^{5} T^{21} + 21818044 p^{6} T^{22} + 5232840 p^{7} T^{23} + 1031295 p^{8} T^{24} + 234144 p^{9} T^{25} + 44737 p^{10} T^{26} + 9121 p^{11} T^{27} + 1645 p^{12} T^{28} + 265 p^{13} T^{29} + 54 p^{14} T^{30} + 7 p^{15} T^{31} + p^{16} T^{32} \)
23 \( 1 - 11 T + 48 T^{2} - 65 T^{3} + 550 T^{4} - 1778 T^{5} - 26167 T^{6} + 201426 T^{7} - 65685 T^{8} + 103885 T^{9} - 18882988 T^{10} + 55884933 T^{11} + 352223981 T^{12} - 607005705 T^{13} - 7507624060 T^{14} - 3921543469 T^{15} + 265495282439 T^{16} - 3921543469 p T^{17} - 7507624060 p^{2} T^{18} - 607005705 p^{3} T^{19} + 352223981 p^{4} T^{20} + 55884933 p^{5} T^{21} - 18882988 p^{6} T^{22} + 103885 p^{7} T^{23} - 65685 p^{8} T^{24} + 201426 p^{9} T^{25} - 26167 p^{10} T^{26} - 1778 p^{11} T^{27} + 550 p^{12} T^{28} - 65 p^{13} T^{29} + 48 p^{14} T^{30} - 11 p^{15} T^{31} + p^{16} T^{32} \)
29 \( 1 - 23 T + 204 T^{2} - 920 T^{3} + 4450 T^{4} - 41279 T^{5} + 256487 T^{6} - 845841 T^{7} + 3746370 T^{8} - 1143295 p T^{9} + 154862819 T^{10} + 91819893 T^{11} - 4498399624 T^{12} + 18513302685 T^{13} - 45067117045 T^{14} + 22912322194 p T^{15} - 5859128371783 T^{16} + 22912322194 p^{2} T^{17} - 45067117045 p^{2} T^{18} + 18513302685 p^{3} T^{19} - 4498399624 p^{4} T^{20} + 91819893 p^{5} T^{21} + 154862819 p^{6} T^{22} - 1143295 p^{8} T^{23} + 3746370 p^{8} T^{24} - 845841 p^{9} T^{25} + 256487 p^{10} T^{26} - 41279 p^{11} T^{27} + 4450 p^{12} T^{28} - 920 p^{13} T^{29} + 204 p^{14} T^{30} - 23 p^{15} T^{31} + p^{16} T^{32} \)
37 \( 1 - 17 T + 40 T^{2} + 13 p T^{3} - 21 T^{4} - 5918 T^{5} - 127075 T^{6} - 13193 T^{7} + 3498407 T^{8} + 11297512 T^{9} - 32502765 T^{10} + 249272800 T^{11} + 2513737506 T^{12} - 55111609620 T^{13} - 203183391261 T^{14} + 848215098021 T^{15} + 14743507216285 T^{16} + 848215098021 p T^{17} - 203183391261 p^{2} T^{18} - 55111609620 p^{3} T^{19} + 2513737506 p^{4} T^{20} + 249272800 p^{5} T^{21} - 32502765 p^{6} T^{22} + 11297512 p^{7} T^{23} + 3498407 p^{8} T^{24} - 13193 p^{9} T^{25} - 127075 p^{10} T^{26} - 5918 p^{11} T^{27} - 21 p^{12} T^{28} + 13 p^{14} T^{29} + 40 p^{14} T^{30} - 17 p^{15} T^{31} + p^{16} T^{32} \)
41 \( 1 - 8 T + 163 T^{2} - 224 T^{3} + 8009 T^{4} + 42535 T^{5} + 545559 T^{6} + 2581183 T^{7} + 46372828 T^{8} + 190553267 T^{9} + 2457333412 T^{10} + 13309651136 T^{11} + 138222477370 T^{12} + 612531577631 T^{13} + 6740220902839 T^{14} + 32378874303626 T^{15} + 263498778072051 T^{16} + 32378874303626 p T^{17} + 6740220902839 p^{2} T^{18} + 612531577631 p^{3} T^{19} + 138222477370 p^{4} T^{20} + 13309651136 p^{5} T^{21} + 2457333412 p^{6} T^{22} + 190553267 p^{7} T^{23} + 46372828 p^{8} T^{24} + 2581183 p^{9} T^{25} + 545559 p^{10} T^{26} + 42535 p^{11} T^{27} + 8009 p^{12} T^{28} - 224 p^{13} T^{29} + 163 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \)
43 \( 1 - 32 T + 567 T^{2} - 7047 T^{3} + 67746 T^{4} - 515905 T^{5} + 3024605 T^{6} - 11699016 T^{7} + 3725142 T^{8} + 397745987 T^{9} - 4063071951 T^{10} + 26896093315 T^{11} - 144867230793 T^{12} + 784523745393 T^{13} - 5146681522601 T^{14} + 38435968017387 T^{15} - 268655730859831 T^{16} + 38435968017387 p T^{17} - 5146681522601 p^{2} T^{18} + 784523745393 p^{3} T^{19} - 144867230793 p^{4} T^{20} + 26896093315 p^{5} T^{21} - 4063071951 p^{6} T^{22} + 397745987 p^{7} T^{23} + 3725142 p^{8} T^{24} - 11699016 p^{9} T^{25} + 3024605 p^{10} T^{26} - 515905 p^{11} T^{27} + 67746 p^{12} T^{28} - 7047 p^{13} T^{29} + 567 p^{14} T^{30} - 32 p^{15} T^{31} + p^{16} T^{32} \)
47 \( 1 + 7 T - 53 T^{2} + 550 T^{3} + 7380 T^{4} - 36299 T^{5} + 71747 T^{6} + 2821792 T^{7} - 13122605 T^{8} + 15027625 T^{9} + 545409133 T^{10} - 6651602549 T^{11} + 3725421441 T^{12} + 169760059900 T^{13} - 53924786750 p T^{14} - 8081152935338 T^{15} + 73752406210659 T^{16} - 8081152935338 p T^{17} - 53924786750 p^{3} T^{18} + 169760059900 p^{3} T^{19} + 3725421441 p^{4} T^{20} - 6651602549 p^{5} T^{21} + 545409133 p^{6} T^{22} + 15027625 p^{7} T^{23} - 13122605 p^{8} T^{24} + 2821792 p^{9} T^{25} + 71747 p^{10} T^{26} - 36299 p^{11} T^{27} + 7380 p^{12} T^{28} + 550 p^{13} T^{29} - 53 p^{14} T^{30} + 7 p^{15} T^{31} + p^{16} T^{32} \)
53 \( 1 - 54 T + 1580 T^{2} - 31962 T^{3} + 498059 T^{4} - 6354441 T^{5} + 69327320 T^{6} - 668215434 T^{7} + 5845905277 T^{8} - 47662996791 T^{9} + 374430447050 T^{10} - 2948021790525 T^{11} + 23919130956286 T^{12} - 198917348958405 T^{13} + 1643259281894031 T^{14} - 13061879929608162 T^{15} + 98098022530226205 T^{16} - 13061879929608162 p T^{17} + 1643259281894031 p^{2} T^{18} - 198917348958405 p^{3} T^{19} + 23919130956286 p^{4} T^{20} - 2948021790525 p^{5} T^{21} + 374430447050 p^{6} T^{22} - 47662996791 p^{7} T^{23} + 5845905277 p^{8} T^{24} - 668215434 p^{9} T^{25} + 69327320 p^{10} T^{26} - 6354441 p^{11} T^{27} + 498059 p^{12} T^{28} - 31962 p^{13} T^{29} + 1580 p^{14} T^{30} - 54 p^{15} T^{31} + p^{16} T^{32} \)
59 \( 1 + 38 T + 574 T^{2} + 3060 T^{3} - 25925 T^{4} - 491816 T^{5} - 1361093 T^{6} + 35571036 T^{7} + 449890110 T^{8} + 1531576300 T^{9} - 13606084901 T^{10} - 159486713323 T^{11} - 69528724639 T^{12} + 9961181826550 T^{13} + 74166930878535 T^{14} - 12586929886941 T^{15} - 2804819675730703 T^{16} - 12586929886941 p T^{17} + 74166930878535 p^{2} T^{18} + 9961181826550 p^{3} T^{19} - 69528724639 p^{4} T^{20} - 159486713323 p^{5} T^{21} - 13606084901 p^{6} T^{22} + 1531576300 p^{7} T^{23} + 449890110 p^{8} T^{24} + 35571036 p^{9} T^{25} - 1361093 p^{10} T^{26} - 491816 p^{11} T^{27} - 25925 p^{12} T^{28} + 3060 p^{13} T^{29} + 574 p^{14} T^{30} + 38 p^{15} T^{31} + p^{16} T^{32} \)
61 \( ( 1 + 14 T + 295 T^{2} + 2920 T^{3} + 41195 T^{4} + 352967 T^{5} + 3965078 T^{6} + 28806460 T^{7} + 274235785 T^{8} + 28806460 p T^{9} + 3965078 p^{2} T^{10} + 352967 p^{3} T^{11} + 41195 p^{4} T^{12} + 2920 p^{5} T^{13} + 295 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
67 \( 1 - T - 212 T^{2} + 1391 T^{3} + 16401 T^{4} - 256360 T^{5} - 220480 T^{6} + 20357228 T^{7} - 50675443 T^{8} - 1134207799 T^{9} + 5342539926 T^{10} + 72580986170 T^{11} - 551327569218 T^{12} - 4042002852936 T^{13} + 52312586283531 T^{14} + 102931143613089 T^{15} - 3834386676959081 T^{16} + 102931143613089 p T^{17} + 52312586283531 p^{2} T^{18} - 4042002852936 p^{3} T^{19} - 551327569218 p^{4} T^{20} + 72580986170 p^{5} T^{21} + 5342539926 p^{6} T^{22} - 1134207799 p^{7} T^{23} - 50675443 p^{8} T^{24} + 20357228 p^{9} T^{25} - 220480 p^{10} T^{26} - 256360 p^{11} T^{27} + 16401 p^{12} T^{28} + 1391 p^{13} T^{29} - 212 p^{14} T^{30} - p^{15} T^{31} + p^{16} T^{32} \)
71 \( 1 + 11 T + 56 T^{2} - 430 T^{3} - 2235 T^{4} - 53947 T^{5} - 153702 T^{6} - 644367 T^{7} + 30915610 T^{8} - 111342765 T^{9} + 713405921 T^{10} - 14050970059 T^{11} - 70933385389 T^{12} - 2228986257475 T^{13} + 9428031356815 T^{14} + 82258584774332 T^{15} + 1926254947121287 T^{16} + 82258584774332 p T^{17} + 9428031356815 p^{2} T^{18} - 2228986257475 p^{3} T^{19} - 70933385389 p^{4} T^{20} - 14050970059 p^{5} T^{21} + 713405921 p^{6} T^{22} - 111342765 p^{7} T^{23} + 30915610 p^{8} T^{24} - 644367 p^{9} T^{25} - 153702 p^{10} T^{26} - 53947 p^{11} T^{27} - 2235 p^{12} T^{28} - 430 p^{13} T^{29} + 56 p^{14} T^{30} + 11 p^{15} T^{31} + p^{16} T^{32} \)
73 \( 1 - 53 T + 1437 T^{2} - 25553 T^{3} + 318796 T^{4} - 2648585 T^{5} + 8560270 T^{6} + 142677066 T^{7} - 3008094258 T^{8} + 30763065723 T^{9} - 176556012296 T^{10} + 13885533175 T^{11} + 11453280633577 T^{12} - 126817968906918 T^{13} + 662719896196834 T^{14} - 20868952201627 T^{15} - 22477057860482081 T^{16} - 20868952201627 p T^{17} + 662719896196834 p^{2} T^{18} - 126817968906918 p^{3} T^{19} + 11453280633577 p^{4} T^{20} + 13885533175 p^{5} T^{21} - 176556012296 p^{6} T^{22} + 30763065723 p^{7} T^{23} - 3008094258 p^{8} T^{24} + 142677066 p^{9} T^{25} + 8560270 p^{10} T^{26} - 2648585 p^{11} T^{27} + 318796 p^{12} T^{28} - 25553 p^{13} T^{29} + 1437 p^{14} T^{30} - 53 p^{15} T^{31} + p^{16} T^{32} \)
79 \( 1 - 48 T + 1184 T^{2} - 20835 T^{3} + 281070 T^{4} - 2904829 T^{5} + 22720462 T^{6} - 120648471 T^{7} + 271602190 T^{8} + 343493350 T^{9} + 24614009439 T^{10} - 701631065922 T^{11} + 8309180969701 T^{12} - 37968691736605 T^{13} - 383983383157800 T^{14} + 10080917517717466 T^{15} - 115661859324573623 T^{16} + 10080917517717466 p T^{17} - 383983383157800 p^{2} T^{18} - 37968691736605 p^{3} T^{19} + 8309180969701 p^{4} T^{20} - 701631065922 p^{5} T^{21} + 24614009439 p^{6} T^{22} + 343493350 p^{7} T^{23} + 271602190 p^{8} T^{24} - 120648471 p^{9} T^{25} + 22720462 p^{10} T^{26} - 2904829 p^{11} T^{27} + 281070 p^{12} T^{28} - 20835 p^{13} T^{29} + 1184 p^{14} T^{30} - 48 p^{15} T^{31} + p^{16} T^{32} \)
83 \( 1 - 56 T + 1835 T^{2} - 42653 T^{3} + 770844 T^{4} - 11130524 T^{5} + 128948895 T^{6} - 1146610071 T^{7} + 6517124257 T^{8} + 6347602281 T^{9} - 735874278340 T^{10} + 11046415448635 T^{11} - 104035840379404 T^{12} + 617025166268050 T^{13} - 638584732911899 T^{14} - 35468513499895598 T^{15} + 474193140439843975 T^{16} - 35468513499895598 p T^{17} - 638584732911899 p^{2} T^{18} + 617025166268050 p^{3} T^{19} - 104035840379404 p^{4} T^{20} + 11046415448635 p^{5} T^{21} - 735874278340 p^{6} T^{22} + 6347602281 p^{7} T^{23} + 6517124257 p^{8} T^{24} - 1146610071 p^{9} T^{25} + 128948895 p^{10} T^{26} - 11130524 p^{11} T^{27} + 770844 p^{12} T^{28} - 42653 p^{13} T^{29} + 1835 p^{14} T^{30} - 56 p^{15} T^{31} + p^{16} T^{32} \)
89 \( 1 + 43 T + 594 T^{2} - 620 T^{3} - 101705 T^{4} - 8764 p T^{5} + 8237282 T^{6} + 185900811 T^{7} + 748317915 T^{8} - 12318825125 T^{9} - 158303894266 T^{10} - 87925790313 T^{11} + 12674280524666 T^{12} + 116376731685960 T^{13} + 186498867435350 T^{14} - 5209874980651966 T^{15} - 67917197270582893 T^{16} - 5209874980651966 p T^{17} + 186498867435350 p^{2} T^{18} + 116376731685960 p^{3} T^{19} + 12674280524666 p^{4} T^{20} - 87925790313 p^{5} T^{21} - 158303894266 p^{6} T^{22} - 12318825125 p^{7} T^{23} + 748317915 p^{8} T^{24} + 185900811 p^{9} T^{25} + 8237282 p^{10} T^{26} - 8764 p^{12} T^{27} - 101705 p^{12} T^{28} - 620 p^{13} T^{29} + 594 p^{14} T^{30} + 43 p^{15} T^{31} + p^{16} T^{32} \)
97 \( 1 - 17 T - 43 T^{2} + 1855 T^{3} - 6735 T^{4} + 198054 T^{5} - 998023 T^{6} - 48717832 T^{7} + 421608520 T^{8} + 972269505 T^{9} - 2352761852 T^{10} + 96893285304 T^{11} - 4316966334749 T^{12} + 16728212134365 T^{13} + 270316350494950 T^{14} - 2104813575661822 T^{15} + 8423896910168359 T^{16} - 2104813575661822 p T^{17} + 270316350494950 p^{2} T^{18} + 16728212134365 p^{3} T^{19} - 4316966334749 p^{4} T^{20} + 96893285304 p^{5} T^{21} - 2352761852 p^{6} T^{22} + 972269505 p^{7} T^{23} + 421608520 p^{8} T^{24} - 48717832 p^{9} T^{25} - 998023 p^{10} T^{26} + 198054 p^{11} T^{27} - 6735 p^{12} T^{28} + 1855 p^{13} T^{29} - 43 p^{14} T^{30} - 17 p^{15} T^{31} + p^{16} T^{32} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.41590266597781676075888778138, −2.36894643891019095403698140982, −2.34672924855340673485954841031, −2.27991865555870511251429737008, −2.24772646641833947371693439049, −2.23789801943229518223080977394, −2.21486501768234558000902186837, −2.17922813291680362061844909552, −2.16236590343910988493840770600, −2.12830462015655258779106104086, −1.81060648204519895512584246683, −1.64634536697223679807423170898, −1.60762186808791676071564877858, −1.57610186925334770196605308193, −1.45381808967758181751740830936, −1.35277787578239339995158470296, −1.18629059666270444069200919607, −1.00921050164070118481109850640, −0.974272404381085588304838325935, −0.933556858038732504443952175782, −0.67514818138897755827136116777, −0.58766490202990709272815771990, −0.46094623808724837476550820395, −0.44079399383126228847166121742, −0.35033497160388401345084697559, 0.35033497160388401345084697559, 0.44079399383126228847166121742, 0.46094623808724837476550820395, 0.58766490202990709272815771990, 0.67514818138897755827136116777, 0.933556858038732504443952175782, 0.974272404381085588304838325935, 1.00921050164070118481109850640, 1.18629059666270444069200919607, 1.35277787578239339995158470296, 1.45381808967758181751740830936, 1.57610186925334770196605308193, 1.60762186808791676071564877858, 1.64634536697223679807423170898, 1.81060648204519895512584246683, 2.12830462015655258779106104086, 2.16236590343910988493840770600, 2.17922813291680362061844909552, 2.21486501768234558000902186837, 2.23789801943229518223080977394, 2.24772646641833947371693439049, 2.27991865555870511251429737008, 2.34672924855340673485954841031, 2.36894643891019095403698140982, 2.41590266597781676075888778138

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

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