Dirichlet series
| L(s) = 1 | + 4·3-s − 5·4-s + 14·9-s − 20·12-s + 12·13-s + 16·16-s − 8·17-s − 12·23-s − 20·25-s + 32·27-s − 16·29-s − 70·36-s + 48·39-s + 16·43-s + 64·48-s − 32·51-s − 60·52-s − 20·53-s + 12·61-s − 25·64-s + 40·68-s − 48·69-s − 80·75-s + 20·79-s + 69·81-s − 64·87-s + 60·92-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 5/2·4-s + 14/3·9-s − 5.77·12-s + 3.32·13-s + 4·16-s − 1.94·17-s − 2.50·23-s − 4·25-s + 6.15·27-s − 2.97·29-s − 11.6·36-s + 7.68·39-s + 2.43·43-s + 9.23·48-s − 4.48·51-s − 8.32·52-s − 2.74·53-s + 1.53·61-s − 3.12·64-s + 4.85·68-s − 5.77·69-s − 9.23·75-s + 2.25·79-s + 23/3·81-s − 6.86·87-s + 6.25·92-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{32} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{32} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(7^{32} \cdot 13^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.00754\times 10^{11}\) |
| Root analytic conductor: | \(2.25532\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 7^{32} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(8.183442291\) |
| \(L(\frac12)\) | \(\approx\) | \(8.183442291\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 \) |
| 13 | \( ( 1 - 6 T + 28 T^{2} - 10 p T^{3} + 46 p T^{4} - 10 p^{2} T^{5} + 28 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| good | 2 | \( 1 + 5 T^{2} + 9 T^{4} - 5 p T^{6} - 9 p^{3} T^{8} - 65 p T^{10} - 11 T^{12} + 475 T^{14} + 1357 T^{16} + 475 p^{2} T^{18} - 11 p^{4} T^{20} - 65 p^{7} T^{22} - 9 p^{11} T^{24} - 5 p^{11} T^{26} + 9 p^{12} T^{28} + 5 p^{14} T^{30} + p^{16} T^{32} \) |
| 3 | \( ( 1 - 2 T - T^{2} + 2 p T^{3} - 5 T^{4} - 8 T^{5} + 4 p^{2} T^{6} - 16 T^{7} - 86 T^{8} - 16 p T^{9} + 4 p^{4} T^{10} - 8 p^{3} T^{11} - 5 p^{4} T^{12} + 2 p^{6} T^{13} - p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 5 | \( 1 + 4 p T^{2} + 197 T^{4} + 252 p T^{6} + 5561 T^{8} + 2656 p T^{10} - 38794 T^{12} - 26112 p^{2} T^{14} - 4154346 T^{16} - 26112 p^{4} T^{18} - 38794 p^{4} T^{20} + 2656 p^{7} T^{22} + 5561 p^{8} T^{24} + 252 p^{11} T^{26} + 197 p^{12} T^{28} + 4 p^{15} T^{30} + p^{16} T^{32} \) | |
| 11 | \( 1 + 36 T^{2} + 744 T^{4} + 7792 T^{6} + 21391 T^{8} - 677936 T^{10} - 10026872 T^{12} - 54252292 T^{14} - 158665504 T^{16} - 54252292 p^{2} T^{18} - 10026872 p^{4} T^{20} - 677936 p^{6} T^{22} + 21391 p^{8} T^{24} + 7792 p^{10} T^{26} + 744 p^{12} T^{28} + 36 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( ( 1 + 4 T - 32 T^{2} - 112 T^{3} + 519 T^{4} + 464 T^{5} - 15984 T^{6} + 5980 T^{7} + 400400 T^{8} + 5980 p T^{9} - 15984 p^{2} T^{10} + 464 p^{3} T^{11} + 519 p^{4} T^{12} - 112 p^{5} T^{13} - 32 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 19 | \( 1 + 108 T^{2} + 6032 T^{4} + 239152 T^{6} + 400893 p T^{8} + 205118480 T^{10} + 4841886560 T^{12} + 103641997748 T^{14} + 2047252246960 T^{16} + 103641997748 p^{2} T^{18} + 4841886560 p^{4} T^{20} + 205118480 p^{6} T^{22} + 400893 p^{9} T^{24} + 239152 p^{10} T^{26} + 6032 p^{12} T^{28} + 108 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( ( 1 + 6 T - 61 T^{2} - 226 T^{3} + 3587 T^{4} + 7704 T^{5} - 119516 T^{6} - 48976 T^{7} + 3411258 T^{8} - 48976 p T^{9} - 119516 p^{2} T^{10} + 7704 p^{3} T^{11} + 3587 p^{4} T^{12} - 226 p^{5} T^{13} - 61 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( ( 1 + 4 T + 53 T^{2} + 140 T^{3} + 2016 T^{4} + 140 p T^{5} + 53 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 31 | \( 1 + 168 T^{2} + 455 p T^{4} + 862216 T^{6} + 44662197 T^{8} + 2021816960 T^{10} + 80664529898 T^{12} + 2887780533296 T^{14} + 93847547388382 T^{16} + 2887780533296 p^{2} T^{18} + 80664529898 p^{4} T^{20} + 2021816960 p^{6} T^{22} + 44662197 p^{8} T^{24} + 862216 p^{10} T^{26} + 455 p^{13} T^{28} + 168 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( 1 + 176 T^{2} + 14937 T^{4} + 890560 T^{6} + 45536021 T^{8} + 2150670720 T^{10} + 92423369002 T^{12} + 3573991527920 T^{14} + 131871000743694 T^{16} + 3573991527920 p^{2} T^{18} + 92423369002 p^{4} T^{20} + 2150670720 p^{6} T^{22} + 45536021 p^{8} T^{24} + 890560 p^{10} T^{26} + 14937 p^{12} T^{28} + 176 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( ( 1 - 196 T^{2} + 20328 T^{4} - 1383052 T^{6} + 66725070 T^{8} - 1383052 p^{2} T^{10} + 20328 p^{4} T^{12} - 196 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 4 T + 106 T^{2} - 672 T^{3} + 5314 T^{4} - 672 p T^{5} + 106 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 47 | \( 1 + 180 T^{2} + 13608 T^{4} + 639376 T^{6} + 32488927 T^{8} + 2249843872 T^{10} + 133121236744 T^{12} + 5746119520412 T^{14} + 236856173717408 T^{16} + 5746119520412 p^{2} T^{18} + 133121236744 p^{4} T^{20} + 2249843872 p^{6} T^{22} + 32488927 p^{8} T^{24} + 639376 p^{10} T^{26} + 13608 p^{12} T^{28} + 180 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( ( 1 + 10 T - 112 T^{2} - 800 T^{3} + 13403 T^{4} + 48080 T^{5} - 1042676 T^{6} - 631410 T^{7} + 69659904 T^{8} - 631410 p T^{9} - 1042676 p^{2} T^{10} + 48080 p^{3} T^{11} + 13403 p^{4} T^{12} - 800 p^{5} T^{13} - 112 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 59 | \( 1 + 284 T^{2} + 39264 T^{4} + 3764528 T^{6} + 302785671 T^{8} + 22620736976 T^{10} + 1599801428368 T^{12} + 105068803074052 T^{14} + 6404562569090416 T^{16} + 105068803074052 p^{2} T^{18} + 1599801428368 p^{4} T^{20} + 22620736976 p^{6} T^{22} + 302785671 p^{8} T^{24} + 3764528 p^{10} T^{26} + 39264 p^{12} T^{28} + 284 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( ( 1 - 6 T - 184 T^{2} + 688 T^{3} + 22755 T^{4} - 46672 T^{5} - 2038828 T^{6} + 974270 T^{7} + 145895056 T^{8} + 974270 p T^{9} - 2038828 p^{2} T^{10} - 46672 p^{3} T^{11} + 22755 p^{4} T^{12} + 688 p^{5} T^{13} - 184 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 + 252 T^{2} + 30328 T^{4} + 1886000 T^{6} + 37411311 T^{8} - 2795291712 T^{10} - 55584072648 T^{12} + 28154988826740 T^{14} + 3102055482356320 T^{16} + 28154988826740 p^{2} T^{18} - 55584072648 p^{4} T^{20} - 2795291712 p^{6} T^{22} + 37411311 p^{8} T^{24} + 1886000 p^{10} T^{26} + 30328 p^{12} T^{28} + 252 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 - 276 T^{2} + 36585 T^{4} - 3341788 T^{6} + 252824924 T^{8} - 3341788 p^{2} T^{10} + 36585 p^{4} T^{12} - 276 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 + 324 T^{2} + 49669 T^{4} + 5194844 T^{6} + 459848313 T^{8} + 37402952928 T^{10} + 2900586483030 T^{12} + 228122212408128 T^{14} + 17430061419064966 T^{16} + 228122212408128 p^{2} T^{18} + 2900586483030 p^{4} T^{20} + 37402952928 p^{6} T^{22} + 459848313 p^{8} T^{24} + 5194844 p^{10} T^{26} + 49669 p^{12} T^{28} + 324 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 - 10 T - 225 T^{2} + 1370 T^{3} + 43665 T^{4} - 156040 T^{5} - 5061650 T^{6} + 3638260 T^{7} + 488999794 T^{8} + 3638260 p T^{9} - 5061650 p^{2} T^{10} - 156040 p^{3} T^{11} + 43665 p^{4} T^{12} + 1370 p^{5} T^{13} - 225 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( ( 1 - 368 T^{2} + 74176 T^{4} - 9998736 T^{6} + 969195550 T^{8} - 9998736 p^{2} T^{10} + 74176 p^{4} T^{12} - 368 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 + 272 T^{2} + 24257 T^{4} + 219936 T^{6} + 1101701 T^{8} + 23353490432 T^{10} + 3243938609402 T^{12} + 147596300662320 T^{14} + 2214719190375342 T^{16} + 147596300662320 p^{2} T^{18} + 3243938609402 p^{4} T^{20} + 23353490432 p^{6} T^{22} + 1101701 p^{8} T^{24} + 219936 p^{10} T^{26} + 24257 p^{12} T^{28} + 272 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( ( 1 - 672 T^{2} + 205664 T^{4} - 37489952 T^{6} + 4452439678 T^{8} - 37489952 p^{2} T^{10} + 205664 p^{4} T^{12} - 672 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
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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
−2.88901412124858777108942404710, −2.75039315023373836259544244855, −2.65083657567708775859548676348, −2.51679429682294839959998485319, −2.47982360105324661211541025532, −2.47392773978550753491210806465, −2.25881234113188706505055354902, −2.17144695090236116097738263453, −2.10570375342863579190455259472, −2.05329471131150428611193130751, −1.99073641466202968420150980223, −1.86666090065912908482474729187, −1.76585602550042000802632089459, −1.67005867289457279832749202737, −1.66168352670988500602582882024, −1.38036835468886169391786320184, −1.37212631195397430665609023107, −1.33353836715614200957291215816, −1.20314495725092171805013672431, −1.17691523541677055112906520354, −0.942791724917145975217714649004, −0.55566132545915317747790385286, −0.41386126009867562531824152483, −0.31721537884080758189298760340, −0.30235858155446112711627558615, 0.30235858155446112711627558615, 0.31721537884080758189298760340, 0.41386126009867562531824152483, 0.55566132545915317747790385286, 0.942791724917145975217714649004, 1.17691523541677055112906520354, 1.20314495725092171805013672431, 1.33353836715614200957291215816, 1.37212631195397430665609023107, 1.38036835468886169391786320184, 1.66168352670988500602582882024, 1.67005867289457279832749202737, 1.76585602550042000802632089459, 1.86666090065912908482474729187, 1.99073641466202968420150980223, 2.05329471131150428611193130751, 2.10570375342863579190455259472, 2.17144695090236116097738263453, 2.25881234113188706505055354902, 2.47392773978550753491210806465, 2.47982360105324661211541025532, 2.51679429682294839959998485319, 2.65083657567708775859548676348, 2.75039315023373836259544244855, 2.88901412124858777108942404710
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.