Dirichlet series
| L(s) = 1 | − 4·3-s + 5·4-s + 14·9-s − 20·12-s + 16·16-s − 8·17-s + 12·23-s + 20·25-s − 32·27-s − 16·29-s + 70·36-s − 16·43-s − 64·48-s − 20·49-s + 32·51-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 + 100·100-s + 8·101-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 5/2·4-s + 14/3·9-s − 5.77·12-s + 4·16-s − 1.94·17-s + 2.50·23-s + 4·25-s − 6.15·27-s − 2.97·29-s + 35/3·36-s − 2.43·43-s − 9.23·48-s − 2.85·49-s + 4.48·51-s − 2.74·53-s − 1.53·61-s + 25/8·64-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 + 10·100-s + 0.796·101-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 13^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 13^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(7^{16} \cdot 13^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.01966\times 10^{15}\) |
| Root analytic conductor: | \(3.07348\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 7^{16} \cdot 13^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.039488139\) |
| \(L(\frac12)\) | \(\approx\) | \(2.039488139\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 + 20 T^{2} + 31 p T^{4} + 1612 T^{6} + 10652 T^{8} + 1612 p^{2} T^{10} + 31 p^{5} T^{12} + 20 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 \) | |
| 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 | \( ( 1 + 672 T^{2} + 205664 T^{4} + 37489952 T^{6} + 4452439678 T^{8} + 37489952 p^{2} T^{10} + 205664 p^{4} T^{12} + 672 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.43388833711818713027116516825, −2.32245602013049890425339628067, −2.27787710121572428062245311322, −2.24684739482886339116995408781, −2.23012306559147992797707403618, −2.11426438558131649232679157354, −2.03965221696585184014096007751, −1.88297486101662372058179760125, −1.69210971805597891596114822090, −1.69140306602728976587835404955, −1.62299104869001517069880590976, −1.55234914789366846970366344476, −1.53007624959230023733890985424, −1.52694886330486352667732215955, −1.41070396991921967163025636256, −1.38553577103939756138408292923, −1.28820719151761349354493384758, −1.03550794899203542288566806224, −0.805419249869439120231882774708, −0.798037528409199423036396436205, −0.67609476700361795691577898453, −0.58395785825580217037634592343, −0.51260562632559463837431942590, −0.37874756908028885619114796342, −0.05527137408281841379879947566, 0.05527137408281841379879947566, 0.37874756908028885619114796342, 0.51260562632559463837431942590, 0.58395785825580217037634592343, 0.67609476700361795691577898453, 0.798037528409199423036396436205, 0.805419249869439120231882774708, 1.03550794899203542288566806224, 1.28820719151761349354493384758, 1.38553577103939756138408292923, 1.41070396991921967163025636256, 1.52694886330486352667732215955, 1.53007624959230023733890985424, 1.55234914789366846970366344476, 1.62299104869001517069880590976, 1.69140306602728976587835404955, 1.69210971805597891596114822090, 1.88297486101662372058179760125, 2.03965221696585184014096007751, 2.11426438558131649232679157354, 2.23012306559147992797707403618, 2.24684739482886339116995408781, 2.27787710121572428062245311322, 2.32245602013049890425339628067, 2.43388833711818713027116516825
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.