Properties

Label 40-370e20-1.1-c1e20-0-1
Degree $40$
Conductor $2.312\times 10^{51}$
Sign $1$
Analytic cond. $2.56789\times 10^{9}$
Root an. cond. $1.71885$
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  − 20·2-s + 4·3-s + 210·4-s − 4·5-s − 80·6-s − 2·7-s − 1.54e3·8-s + 8·9-s + 80·10-s + 840·12-s + 40·14-s − 16·15-s + 8.85e3·16-s − 160·18-s + 6·19-s − 840·20-s − 8·21-s − 4·23-s − 6.16e3·24-s + 13·25-s + 8·27-s − 420·28-s + 18·29-s + 320·30-s + 12·31-s − 4.25e4·32-s + 8·35-s + ⋯
L(s)  = 1  − 14.1·2-s + 2.30·3-s + 105·4-s − 1.78·5-s − 32.6·6-s − 0.755·7-s − 544.·8-s + 8/3·9-s + 25.2·10-s + 242.·12-s + 10.6·14-s − 4.13·15-s + 2.21e3·16-s − 37.7·18-s + 1.37·19-s − 187.·20-s − 1.74·21-s − 0.834·23-s − 1.25e3·24-s + 13/5·25-s + 1.53·27-s − 79.3·28-s + 3.34·29-s + 58.4·30-s + 2.15·31-s − 7.51e3·32-s + 1.35·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{20} \cdot 37^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{20} \cdot 37^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(2^{20} \cdot 5^{20} \cdot 37^{20}\)
Sign: $1$
Analytic conductor: \(2.56789\times 10^{9}\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{370} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 2^{20} \cdot 5^{20} \cdot 37^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.004671557052\)
\(L(\frac12)\) \(\approx\) \(0.004671557052\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T )^{20} \)
5 \( 1 + 4 T + 3 T^{2} + 28 T^{4} + 12 p T^{5} - 116 T^{6} - 204 T^{7} + 19 p^{2} T^{8} - 1956 T^{9} - 11342 T^{10} - 1956 p T^{11} + 19 p^{4} T^{12} - 204 p^{3} T^{13} - 116 p^{4} T^{14} + 12 p^{6} T^{15} + 28 p^{6} T^{16} + 3 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \)
37 \( 1 + 32 T + 492 T^{2} + 4288 T^{3} + 15789 T^{4} - 109600 T^{5} - 2041472 T^{6} - 13421280 T^{7} - 564574 p T^{8} + 407467424 T^{9} + 4017596840 T^{10} + 407467424 p T^{11} - 564574 p^{3} T^{12} - 13421280 p^{3} T^{13} - 2041472 p^{4} T^{14} - 109600 p^{5} T^{15} + 15789 p^{6} T^{16} + 4288 p^{7} T^{17} + 492 p^{8} T^{18} + 32 p^{9} T^{19} + p^{10} T^{20} \)
good3 \( 1 - 4 T + 8 T^{2} - 8 T^{3} - 11 T^{4} + 62 T^{5} - 128 T^{6} + 116 T^{7} + 146 T^{8} - 274 p T^{9} + 1706 T^{10} - 692 p T^{11} + 1202 T^{12} + 1856 T^{13} - 7214 T^{14} + 15374 T^{15} - 21203 T^{16} + 9940 T^{17} + 44396 T^{18} - 177670 T^{19} + 369970 T^{20} - 177670 p T^{21} + 44396 p^{2} T^{22} + 9940 p^{3} T^{23} - 21203 p^{4} T^{24} + 15374 p^{5} T^{25} - 7214 p^{6} T^{26} + 1856 p^{7} T^{27} + 1202 p^{8} T^{28} - 692 p^{10} T^{29} + 1706 p^{10} T^{30} - 274 p^{12} T^{31} + 146 p^{12} T^{32} + 116 p^{13} T^{33} - 128 p^{14} T^{34} + 62 p^{15} T^{35} - 11 p^{16} T^{36} - 8 p^{17} T^{37} + 8 p^{18} T^{38} - 4 p^{19} T^{39} + p^{20} T^{40} \)
7 \( 1 + 2 T + 2 T^{2} + 2 p T^{3} - 13 T^{4} + 92 T^{5} + 44 p T^{6} + 788 T^{7} + 2433 T^{8} + 2942 T^{9} + 414 p^{2} T^{10} + 18402 T^{11} - 18980 T^{12} - 106 p T^{13} + 389594 T^{14} - 32602 T^{15} + 7367070 T^{16} + 5570858 T^{17} - 484366 p^{2} T^{18} + 3205786 p T^{19} - 347596702 T^{20} + 3205786 p^{2} T^{21} - 484366 p^{4} T^{22} + 5570858 p^{3} T^{23} + 7367070 p^{4} T^{24} - 32602 p^{5} T^{25} + 389594 p^{6} T^{26} - 106 p^{8} T^{27} - 18980 p^{8} T^{28} + 18402 p^{9} T^{29} + 414 p^{12} T^{30} + 2942 p^{11} T^{31} + 2433 p^{12} T^{32} + 788 p^{13} T^{33} + 44 p^{15} T^{34} + 92 p^{15} T^{35} - 13 p^{16} T^{36} + 2 p^{18} T^{37} + 2 p^{18} T^{38} + 2 p^{19} T^{39} + p^{20} T^{40} \)
11 \( 1 - 108 T^{2} + 6246 T^{4} - 250586 T^{6} + 702435 p T^{8} - 193075702 T^{10} + 4034961265 T^{12} - 71942366094 T^{14} + 1108584758210 T^{16} - 14883472017910 T^{18} + 174879069451146 T^{20} - 14883472017910 p^{2} T^{22} + 1108584758210 p^{4} T^{24} - 71942366094 p^{6} T^{26} + 4034961265 p^{8} T^{28} - 193075702 p^{10} T^{30} + 702435 p^{13} T^{32} - 250586 p^{14} T^{34} + 6246 p^{16} T^{36} - 108 p^{18} T^{38} + p^{20} T^{40} \)
13 \( ( 1 + 47 T^{2} + 32 T^{3} + 1068 T^{4} + 1090 T^{5} + 14672 T^{6} + 18750 T^{7} + 126331 T^{8} + 258456 T^{9} + 1043442 T^{10} + 258456 p T^{11} + 126331 p^{2} T^{12} + 18750 p^{3} T^{13} + 14672 p^{4} T^{14} + 1090 p^{5} T^{15} + 1068 p^{6} T^{16} + 32 p^{7} T^{17} + 47 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
17 \( 1 - 126 T^{2} + 7815 T^{4} - 311858 T^{6} + 8864937 T^{8} - 189078532 T^{10} + 3153624604 T^{12} - 44160055452 T^{14} + 600917312726 T^{16} - 9222670559344 T^{18} + 155104504376634 T^{20} - 9222670559344 p^{2} T^{22} + 600917312726 p^{4} T^{24} - 44160055452 p^{6} T^{26} + 3153624604 p^{8} T^{28} - 189078532 p^{10} T^{30} + 8864937 p^{12} T^{32} - 311858 p^{14} T^{34} + 7815 p^{16} T^{36} - 126 p^{18} T^{38} + p^{20} T^{40} \)
19 \( 1 - 6 T + 18 T^{2} + 14 T^{3} - 910 T^{4} + 3666 T^{5} - 5518 T^{6} - 18026 T^{7} + 366381 T^{8} - 1417104 T^{9} + 3353800 T^{10} - 7862448 T^{11} - 54140360 T^{12} + 414417120 T^{13} - 1581037080 T^{14} + 6408910240 T^{15} - 3629532958 T^{16} - 70133937612 T^{17} + 274674638172 T^{18} - 978584443684 T^{19} + 3359099901228 T^{20} - 978584443684 p T^{21} + 274674638172 p^{2} T^{22} - 70133937612 p^{3} T^{23} - 3629532958 p^{4} T^{24} + 6408910240 p^{5} T^{25} - 1581037080 p^{6} T^{26} + 414417120 p^{7} T^{27} - 54140360 p^{8} T^{28} - 7862448 p^{9} T^{29} + 3353800 p^{10} T^{30} - 1417104 p^{11} T^{31} + 366381 p^{12} T^{32} - 18026 p^{13} T^{33} - 5518 p^{14} T^{34} + 3666 p^{15} T^{35} - 910 p^{16} T^{36} + 14 p^{17} T^{37} + 18 p^{18} T^{38} - 6 p^{19} T^{39} + p^{20} T^{40} \)
23 \( ( 1 + 2 T + 5 p T^{2} + 444 T^{3} + 6764 T^{4} + 34986 T^{5} + 291800 T^{6} + 1532394 T^{7} + 10124459 T^{8} + 45851850 T^{9} + 270082106 T^{10} + 45851850 p T^{11} + 10124459 p^{2} T^{12} + 1532394 p^{3} T^{13} + 291800 p^{4} T^{14} + 34986 p^{5} T^{15} + 6764 p^{6} T^{16} + 444 p^{7} T^{17} + 5 p^{9} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
29 \( 1 - 18 T + 162 T^{2} - 1362 T^{3} + 12774 T^{4} - 97956 T^{5} + 621342 T^{6} - 4096542 T^{7} + 26767981 T^{8} - 147030468 T^{9} + 750291966 T^{10} - 4041512496 T^{11} + 19116836065 T^{12} - 68802600192 T^{13} + 223103658582 T^{14} - 457072047264 T^{15} - 4079313694802 T^{16} + 51380228770812 T^{17} - 348451102363572 T^{18} + 2347856733467382 T^{19} - 14428340269140998 T^{20} + 2347856733467382 p T^{21} - 348451102363572 p^{2} T^{22} + 51380228770812 p^{3} T^{23} - 4079313694802 p^{4} T^{24} - 457072047264 p^{5} T^{25} + 223103658582 p^{6} T^{26} - 68802600192 p^{7} T^{27} + 19116836065 p^{8} T^{28} - 4041512496 p^{9} T^{29} + 750291966 p^{10} T^{30} - 147030468 p^{11} T^{31} + 26767981 p^{12} T^{32} - 4096542 p^{13} T^{33} + 621342 p^{14} T^{34} - 97956 p^{15} T^{35} + 12774 p^{16} T^{36} - 1362 p^{17} T^{37} + 162 p^{18} T^{38} - 18 p^{19} T^{39} + p^{20} T^{40} \)
31 \( 1 - 12 T + 72 T^{2} - 356 T^{3} + 842 T^{4} - 2678 T^{5} + 34880 T^{6} - 352488 T^{7} + 3252705 T^{8} - 18452408 T^{9} + 86936746 T^{10} - 446102644 T^{11} + 3173287089 T^{12} - 23534687474 T^{13} + 137636473708 T^{14} - 746291521086 T^{15} + 3204450303850 T^{16} - 15379961302370 T^{17} + 100682544666346 T^{18} - 721087077048368 T^{19} + 4807016938508210 T^{20} - 721087077048368 p T^{21} + 100682544666346 p^{2} T^{22} - 15379961302370 p^{3} T^{23} + 3204450303850 p^{4} T^{24} - 746291521086 p^{5} T^{25} + 137636473708 p^{6} T^{26} - 23534687474 p^{7} T^{27} + 3173287089 p^{8} T^{28} - 446102644 p^{9} T^{29} + 86936746 p^{10} T^{30} - 18452408 p^{11} T^{31} + 3252705 p^{12} T^{32} - 352488 p^{13} T^{33} + 34880 p^{14} T^{34} - 2678 p^{15} T^{35} + 842 p^{16} T^{36} - 356 p^{17} T^{37} + 72 p^{18} T^{38} - 12 p^{19} T^{39} + p^{20} T^{40} \)
41 \( 1 - 352 T^{2} + 61574 T^{4} - 7003054 T^{6} + 568260057 T^{8} - 33681655358 T^{10} + 1387007520881 T^{12} - 28500433650714 T^{14} - 953873260275422 T^{16} + 124908350856883494 T^{18} - 6591784544503261414 T^{20} + 124908350856883494 p^{2} T^{22} - 953873260275422 p^{4} T^{24} - 28500433650714 p^{6} T^{26} + 1387007520881 p^{8} T^{28} - 33681655358 p^{10} T^{30} + 568260057 p^{12} T^{32} - 7003054 p^{14} T^{34} + 61574 p^{16} T^{36} - 352 p^{18} T^{38} + p^{20} T^{40} \)
43 \( ( 1 - 8 T + 209 T^{2} - 2026 T^{3} + 25887 T^{4} - 232358 T^{5} + 2312588 T^{6} - 17713326 T^{7} + 149372440 T^{8} - 1014135210 T^{9} + 7274031046 T^{10} - 1014135210 p T^{11} + 149372440 p^{2} T^{12} - 17713326 p^{3} T^{13} + 2312588 p^{4} T^{14} - 232358 p^{5} T^{15} + 25887 p^{6} T^{16} - 2026 p^{7} T^{17} + 209 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
47 \( 1 + 22 T + 242 T^{2} + 2850 T^{3} + 37242 T^{4} + 358438 T^{5} + 2934322 T^{6} + 27887618 T^{7} + 246271693 T^{8} + 1680777376 T^{9} + 11817943864 T^{10} + 94471485344 T^{11} + 568828685560 T^{12} + 2523781077136 T^{13} + 15660162847288 T^{14} + 83189365412336 T^{15} - 121832291681678 T^{16} - 3359933133422756 T^{17} - 18402746388641876 T^{18} - 236767748206273516 T^{19} - 2459396131898505956 T^{20} - 236767748206273516 p T^{21} - 18402746388641876 p^{2} T^{22} - 3359933133422756 p^{3} T^{23} - 121832291681678 p^{4} T^{24} + 83189365412336 p^{5} T^{25} + 15660162847288 p^{6} T^{26} + 2523781077136 p^{7} T^{27} + 568828685560 p^{8} T^{28} + 94471485344 p^{9} T^{29} + 11817943864 p^{10} T^{30} + 1680777376 p^{11} T^{31} + 246271693 p^{12} T^{32} + 27887618 p^{13} T^{33} + 2934322 p^{14} T^{34} + 358438 p^{15} T^{35} + 37242 p^{16} T^{36} + 2850 p^{17} T^{37} + 242 p^{18} T^{38} + 22 p^{19} T^{39} + p^{20} T^{40} \)
53 \( 1 + 4 T + 8 T^{2} + 244 T^{3} + 8219 T^{4} + 26424 T^{5} + 69712 T^{6} + 1733832 T^{7} + 47124225 T^{8} + 131098908 T^{9} + 346607800 T^{10} + 8240525052 T^{11} + 188985530268 T^{12} + 445206552516 T^{13} + 1112980846312 T^{14} + 26854745251908 T^{15} + 630349805473710 T^{16} + 1368206353814820 T^{17} + 2972097812137288 T^{18} + 75729690856918020 T^{19} + 1861950102177542994 T^{20} + 75729690856918020 p T^{21} + 2972097812137288 p^{2} T^{22} + 1368206353814820 p^{3} T^{23} + 630349805473710 p^{4} T^{24} + 26854745251908 p^{5} T^{25} + 1112980846312 p^{6} T^{26} + 445206552516 p^{7} T^{27} + 188985530268 p^{8} T^{28} + 8240525052 p^{9} T^{29} + 346607800 p^{10} T^{30} + 131098908 p^{11} T^{31} + 47124225 p^{12} T^{32} + 1733832 p^{13} T^{33} + 69712 p^{14} T^{34} + 26424 p^{15} T^{35} + 8219 p^{16} T^{36} + 244 p^{17} T^{37} + 8 p^{18} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \)
59 \( 1 + 10 T + 50 T^{2} + 750 T^{3} - 4518 T^{4} - 113630 T^{5} - 629150 T^{6} - 9767050 T^{7} - 30375683 T^{8} + 587055952 T^{9} + 3677444920 T^{10} + 61181697200 T^{11} + 444477457528 T^{12} - 1537149132032 T^{13} - 13214791082120 T^{14} - 253761679799488 T^{15} - 2948023922769038 T^{16} - 1739855647067372 T^{17} + 20743752877614892 T^{18} + 637954094659884188 T^{19} + 12621622708654139356 T^{20} + 637954094659884188 p T^{21} + 20743752877614892 p^{2} T^{22} - 1739855647067372 p^{3} T^{23} - 2948023922769038 p^{4} T^{24} - 253761679799488 p^{5} T^{25} - 13214791082120 p^{6} T^{26} - 1537149132032 p^{7} T^{27} + 444477457528 p^{8} T^{28} + 61181697200 p^{9} T^{29} + 3677444920 p^{10} T^{30} + 587055952 p^{11} T^{31} - 30375683 p^{12} T^{32} - 9767050 p^{13} T^{33} - 629150 p^{14} T^{34} - 113630 p^{15} T^{35} - 4518 p^{16} T^{36} + 750 p^{17} T^{37} + 50 p^{18} T^{38} + 10 p^{19} T^{39} + p^{20} T^{40} \)
61 \( 1 - 10 T + 50 T^{2} - 1102 T^{3} + 5102 T^{4} + 59504 T^{5} - 242938 T^{6} + 5929502 T^{7} - 118784235 T^{8} + 389703476 T^{9} - 1482301530 T^{10} + 26651068668 T^{11} + 367516403617 T^{12} - 4425202085464 T^{13} + 15980633545190 T^{14} - 312428322858268 T^{15} + 1705812391183350 T^{16} + 9703738660203308 T^{17} - 35710800258586660 T^{18} + 716846940620847082 T^{19} - 13985029345046462326 T^{20} + 716846940620847082 p T^{21} - 35710800258586660 p^{2} T^{22} + 9703738660203308 p^{3} T^{23} + 1705812391183350 p^{4} T^{24} - 312428322858268 p^{5} T^{25} + 15980633545190 p^{6} T^{26} - 4425202085464 p^{7} T^{27} + 367516403617 p^{8} T^{28} + 26651068668 p^{9} T^{29} - 1482301530 p^{10} T^{30} + 389703476 p^{11} T^{31} - 118784235 p^{12} T^{32} + 5929502 p^{13} T^{33} - 242938 p^{14} T^{34} + 59504 p^{15} T^{35} + 5102 p^{16} T^{36} - 1102 p^{17} T^{37} + 50 p^{18} T^{38} - 10 p^{19} T^{39} + p^{20} T^{40} \)
67 \( 1 - 8 T + 32 T^{2} + 488 T^{3} - 6847 T^{4} - 54710 T^{5} + 775856 T^{6} - 9762612 T^{7} + 17977950 T^{8} + 391274494 T^{9} - 1660766926 T^{10} - 2237557496 T^{11} + 476538792886 T^{12} - 3188344635284 T^{13} + 10093685098882 T^{14} + 3393588714954 T^{15} - 603365752730255 T^{16} - 4100495597791448 T^{17} + 55837510583568844 T^{18} - 535879391991961986 T^{19} + 5676916329019635570 T^{20} - 535879391991961986 p T^{21} + 55837510583568844 p^{2} T^{22} - 4100495597791448 p^{3} T^{23} - 603365752730255 p^{4} T^{24} + 3393588714954 p^{5} T^{25} + 10093685098882 p^{6} T^{26} - 3188344635284 p^{7} T^{27} + 476538792886 p^{8} T^{28} - 2237557496 p^{9} T^{29} - 1660766926 p^{10} T^{30} + 391274494 p^{11} T^{31} + 17977950 p^{12} T^{32} - 9762612 p^{13} T^{33} + 775856 p^{14} T^{34} - 54710 p^{15} T^{35} - 6847 p^{16} T^{36} + 488 p^{17} T^{37} + 32 p^{18} T^{38} - 8 p^{19} T^{39} + p^{20} T^{40} \)
71 \( ( 1 - 8 T + 358 T^{2} - 3528 T^{3} + 71837 T^{4} - 714480 T^{5} + 10091912 T^{6} - 93443184 T^{7} + 1059740690 T^{8} - 8807048448 T^{9} + 85572573860 T^{10} - 8807048448 p T^{11} + 1059740690 p^{2} T^{12} - 93443184 p^{3} T^{13} + 10091912 p^{4} T^{14} - 714480 p^{5} T^{15} + 71837 p^{6} T^{16} - 3528 p^{7} T^{17} + 358 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
73 \( 1 + 6 T + 18 T^{2} - 562 T^{3} - 14119 T^{4} - 108894 T^{5} - 241300 T^{6} + 456484 T^{7} + 76026450 T^{8} + 797248254 T^{9} + 5680506334 T^{10} + 41325048144 T^{11} - 34807496942 T^{12} - 3834627960888 T^{13} - 33769021112502 T^{14} - 340280052432398 T^{15} - 2392553004779731 T^{16} - 2111098936325898 T^{17} + 58818750761960634 T^{18} + 1557695701948058312 T^{19} + 21271415429064564522 T^{20} + 1557695701948058312 p T^{21} + 58818750761960634 p^{2} T^{22} - 2111098936325898 p^{3} T^{23} - 2392553004779731 p^{4} T^{24} - 340280052432398 p^{5} T^{25} - 33769021112502 p^{6} T^{26} - 3834627960888 p^{7} T^{27} - 34807496942 p^{8} T^{28} + 41325048144 p^{9} T^{29} + 5680506334 p^{10} T^{30} + 797248254 p^{11} T^{31} + 76026450 p^{12} T^{32} + 456484 p^{13} T^{33} - 241300 p^{14} T^{34} - 108894 p^{15} T^{35} - 14119 p^{16} T^{36} - 562 p^{17} T^{37} + 18 p^{18} T^{38} + 6 p^{19} T^{39} + p^{20} T^{40} \)
79 \( 1 - 12 T + 72 T^{2} + 1112 T^{3} - 8443 T^{4} - 115194 T^{5} + 2608496 T^{6} - 9820532 T^{7} - 128718798 T^{8} + 797212938 T^{9} + 17439127162 T^{10} - 239390403444 T^{11} + 208510209682 T^{12} + 13355234894376 T^{13} - 2089384921230 T^{14} - 1850585672029274 T^{15} + 14611420367070221 T^{16} + 36570001364063700 T^{17} - 735236365087925700 T^{18} - 5114564831154485014 T^{19} + \)\(12\!\cdots\!54\)\( T^{20} - 5114564831154485014 p T^{21} - 735236365087925700 p^{2} T^{22} + 36570001364063700 p^{3} T^{23} + 14611420367070221 p^{4} T^{24} - 1850585672029274 p^{5} T^{25} - 2089384921230 p^{6} T^{26} + 13355234894376 p^{7} T^{27} + 208510209682 p^{8} T^{28} - 239390403444 p^{9} T^{29} + 17439127162 p^{10} T^{30} + 797212938 p^{11} T^{31} - 128718798 p^{12} T^{32} - 9820532 p^{13} T^{33} + 2608496 p^{14} T^{34} - 115194 p^{15} T^{35} - 8443 p^{16} T^{36} + 1112 p^{17} T^{37} + 72 p^{18} T^{38} - 12 p^{19} T^{39} + p^{20} T^{40} \)
83 \( 1 - 6 T + 18 T^{2} - 114 T^{3} - 19414 T^{4} + 142722 T^{5} - 500382 T^{6} + 8638950 T^{7} + 82229277 T^{8} - 1506484128 T^{9} + 7044305976 T^{10} - 151513542432 T^{11} + 833183379000 T^{12} + 9000931297968 T^{13} - 51518837635848 T^{14} + 1146665167608528 T^{15} - 11610303227411214 T^{16} - 20289230745331644 T^{17} + 147936291046807788 T^{18} - 3537898317581946036 T^{19} + 83574848493417194940 T^{20} - 3537898317581946036 p T^{21} + 147936291046807788 p^{2} T^{22} - 20289230745331644 p^{3} T^{23} - 11610303227411214 p^{4} T^{24} + 1146665167608528 p^{5} T^{25} - 51518837635848 p^{6} T^{26} + 9000931297968 p^{7} T^{27} + 833183379000 p^{8} T^{28} - 151513542432 p^{9} T^{29} + 7044305976 p^{10} T^{30} - 1506484128 p^{11} T^{31} + 82229277 p^{12} T^{32} + 8638950 p^{13} T^{33} - 500382 p^{14} T^{34} + 142722 p^{15} T^{35} - 19414 p^{16} T^{36} - 114 p^{17} T^{37} + 18 p^{18} T^{38} - 6 p^{19} T^{39} + p^{20} T^{40} \)
89 \( 1 + 44 T + 968 T^{2} + 14340 T^{3} + 147114 T^{4} + 994100 T^{5} + 4151848 T^{6} + 26693884 T^{7} + 841021741 T^{8} + 17319514784 T^{9} + 214062340096 T^{10} + 1644971664736 T^{11} + 5031538965688 T^{12} - 35630284360000 T^{13} - 259118724244928 T^{14} + 6303883871551168 T^{15} + 153220556960450482 T^{16} + 1590841799833299800 T^{17} + 7749373125946350928 T^{18} - 23707026920525031032 T^{19} - \)\(66\!\cdots\!36\)\( T^{20} - 23707026920525031032 p T^{21} + 7749373125946350928 p^{2} T^{22} + 1590841799833299800 p^{3} T^{23} + 153220556960450482 p^{4} T^{24} + 6303883871551168 p^{5} T^{25} - 259118724244928 p^{6} T^{26} - 35630284360000 p^{7} T^{27} + 5031538965688 p^{8} T^{28} + 1644971664736 p^{9} T^{29} + 214062340096 p^{10} T^{30} + 17319514784 p^{11} T^{31} + 841021741 p^{12} T^{32} + 26693884 p^{13} T^{33} + 4151848 p^{14} T^{34} + 994100 p^{15} T^{35} + 147114 p^{16} T^{36} + 14340 p^{17} T^{37} + 968 p^{18} T^{38} + 44 p^{19} T^{39} + p^{20} T^{40} \)
97 \( 1 - 658 T^{2} + 246359 T^{4} - 66663082 T^{6} + 14424521937 T^{8} - 2620158445040 T^{10} + 412230069570572 T^{12} - 57266359901723904 T^{14} + 7120056935405706862 T^{16} - \)\(79\!\cdots\!32\)\( T^{18} + \)\(81\!\cdots\!98\)\( T^{20} - \)\(79\!\cdots\!32\)\( p^{2} T^{22} + 7120056935405706862 p^{4} T^{24} - 57266359901723904 p^{6} T^{26} + 412230069570572 p^{8} T^{28} - 2620158445040 p^{10} T^{30} + 14424521937 p^{12} T^{32} - 66663082 p^{14} T^{34} + 246359 p^{16} T^{36} - 658 p^{18} T^{38} + p^{20} T^{40} \)
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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

−2.52880547176859144573578115747, −2.40545221324329982598928972411, −2.32921248830963221834218122604, −2.31616812252778997381831312401, −2.26712868570617692916936398062, −2.22615220952263096630196242262, −2.13775012569130910502189786469, −2.01211419774416913897158231476, −1.91667935302090990450380563816, −1.78589529398794047180456442993, −1.75549983936233523907562518504, −1.61756691077382866499540149540, −1.57498701703147038738978322923, −1.56838593941602592229876276462, −1.39077293776433931916942311112, −1.29472393702820553404810635433, −1.13243733653321496779901018892, −1.12908165864844865392877165503, −1.01277652586633846204674401927, −0.865287218671018101826048689564, −0.809098301223123814267765838036, −0.57860011343253862190412420651, −0.43459862122650478939976492554, −0.29745285762675828136349118838, −0.24753968932245616152820951048, 0.24753968932245616152820951048, 0.29745285762675828136349118838, 0.43459862122650478939976492554, 0.57860011343253862190412420651, 0.809098301223123814267765838036, 0.865287218671018101826048689564, 1.01277652586633846204674401927, 1.12908165864844865392877165503, 1.13243733653321496779901018892, 1.29472393702820553404810635433, 1.39077293776433931916942311112, 1.56838593941602592229876276462, 1.57498701703147038738978322923, 1.61756691077382866499540149540, 1.75549983936233523907562518504, 1.78589529398794047180456442993, 1.91667935302090990450380563816, 2.01211419774416913897158231476, 2.13775012569130910502189786469, 2.22615220952263096630196242262, 2.26712868570617692916936398062, 2.31616812252778997381831312401, 2.32921248830963221834218122604, 2.40545221324329982598928972411, 2.52880547176859144573578115747

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

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