Dirichlet series
| L(s) = 1 | − 2·2-s + 4-s − 7·5-s + 12·7-s − 8-s − 6·9-s + 14·10-s + 2·11-s + 9·13-s − 24·14-s − 4·17-s + 12·18-s + 13·19-s − 7·20-s − 4·22-s − 5·23-s + 28·25-s − 18·26-s − 3·27-s + 12·28-s + 7·29-s + 35·31-s + 8·34-s − 84·35-s − 6·36-s + 9·37-s − 26·38-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 3.13·5-s + 4.53·7-s − 0.353·8-s − 2·9-s + 4.42·10-s + 0.603·11-s + 2.49·13-s − 6.41·14-s − 0.970·17-s + 2.82·18-s + 2.98·19-s − 1.56·20-s − 0.852·22-s − 1.04·23-s + 28/5·25-s − 3.53·26-s − 0.577·27-s + 2.26·28-s + 1.29·29-s + 6.28·31-s + 1.37·34-s − 14.1·35-s − 36-s + 1.47·37-s − 4.21·38-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{7} \cdot 61^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{7} \cdot 61^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(5^{7} \cdot 61^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(508.204\) |
| Root analytic conductor: | \(1.56058\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 5^{7} \cdot 61^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.291613590\) |
| \(L(\frac12)\) | \(\approx\) | \(1.291613590\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 + T )^{7} \) |
| 61 | \( ( 1 - T )^{7} \) | |
| good | 2 | \( 1 + p T + 3 T^{2} + 5 T^{3} + 9 T^{4} + p^{4} T^{5} + 25 T^{6} + 29 T^{7} + 25 p T^{8} + p^{6} T^{9} + 9 p^{3} T^{10} + 5 p^{4} T^{11} + 3 p^{5} T^{12} + p^{7} T^{13} + p^{7} T^{14} \) |
| 3 | \( 1 + 2 p T^{2} + p T^{3} + 28 T^{4} + 28 T^{5} + 95 T^{6} + 94 T^{7} + 95 p T^{8} + 28 p^{2} T^{9} + 28 p^{3} T^{10} + p^{5} T^{11} + 2 p^{6} T^{12} + p^{7} T^{14} \) | |
| 7 | \( 1 - 12 T + 82 T^{2} - 403 T^{3} + 1616 T^{4} - 5520 T^{5} + 16711 T^{6} - 45938 T^{7} + 16711 p T^{8} - 5520 p^{2} T^{9} + 1616 p^{3} T^{10} - 403 p^{4} T^{11} + 82 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 - 2 T + 50 T^{2} - 117 T^{3} + 116 p T^{4} - 2818 T^{5} + 21027 T^{6} - 38994 T^{7} + 21027 p T^{8} - 2818 p^{2} T^{9} + 116 p^{4} T^{10} - 117 p^{4} T^{11} + 50 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 9 T + 103 T^{2} - 629 T^{3} + 4201 T^{4} - 19219 T^{5} + 92375 T^{6} - 326494 T^{7} + 92375 p T^{8} - 19219 p^{2} T^{9} + 4201 p^{3} T^{10} - 629 p^{4} T^{11} + 103 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 4 T + 82 T^{2} + 185 T^{3} + 2748 T^{4} + 2968 T^{5} + 57217 T^{6} + 33238 T^{7} + 57217 p T^{8} + 2968 p^{2} T^{9} + 2748 p^{3} T^{10} + 185 p^{4} T^{11} + 82 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - 13 T + 125 T^{2} - 785 T^{3} + 4253 T^{4} - 18095 T^{5} + 77225 T^{6} - 309670 T^{7} + 77225 p T^{8} - 18095 p^{2} T^{9} + 4253 p^{3} T^{10} - 785 p^{4} T^{11} + 125 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 5 T + 89 T^{2} + 341 T^{3} + 3485 T^{4} + 11415 T^{5} + 92597 T^{6} + 286030 T^{7} + 92597 p T^{8} + 11415 p^{2} T^{9} + 3485 p^{3} T^{10} + 341 p^{4} T^{11} + 89 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 7 T + 127 T^{2} - 887 T^{3} + 8425 T^{4} - 52637 T^{5} + 363295 T^{6} - 1899530 T^{7} + 363295 p T^{8} - 52637 p^{2} T^{9} + 8425 p^{3} T^{10} - 887 p^{4} T^{11} + 127 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 35 T + 665 T^{2} - 8775 T^{3} + 90017 T^{4} - 756173 T^{5} + 5351169 T^{6} - 32230598 T^{7} + 5351169 p T^{8} - 756173 p^{2} T^{9} + 90017 p^{3} T^{10} - 8775 p^{4} T^{11} + 665 p^{5} T^{12} - 35 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 9 T + 189 T^{2} - 1573 T^{3} + 17855 T^{4} - 123315 T^{5} + 1037011 T^{6} - 5733054 T^{7} + 1037011 p T^{8} - 123315 p^{2} T^{9} + 17855 p^{3} T^{10} - 1573 p^{4} T^{11} + 189 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 8 T + 168 T^{2} + 879 T^{3} + 13810 T^{4} + 61492 T^{5} + 794317 T^{6} + 2980026 T^{7} + 794317 p T^{8} + 61492 p^{2} T^{9} + 13810 p^{3} T^{10} + 879 p^{4} T^{11} + 168 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 9 T + 169 T^{2} - 1145 T^{3} + 13249 T^{4} - 1969 p T^{5} + 767777 T^{6} - 4478134 T^{7} + 767777 p T^{8} - 1969 p^{3} T^{9} + 13249 p^{3} T^{10} - 1145 p^{4} T^{11} + 169 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 3 T + 145 T^{2} + 381 T^{3} + 11389 T^{4} + 16773 T^{5} + 635181 T^{6} + 621858 T^{7} + 635181 p T^{8} + 16773 p^{2} T^{9} + 11389 p^{3} T^{10} + 381 p^{4} T^{11} + 145 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 5 T + 277 T^{2} + 1055 T^{3} + 35855 T^{4} + 111303 T^{5} + 2872651 T^{6} + 7360474 T^{7} + 2872651 p T^{8} + 111303 p^{2} T^{9} + 35855 p^{3} T^{10} + 1055 p^{4} T^{11} + 277 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 6 T + 234 T^{2} - 727 T^{3} + 25068 T^{4} - 40486 T^{5} + 1901115 T^{6} - 2157062 T^{7} + 1901115 p T^{8} - 40486 p^{2} T^{9} + 25068 p^{3} T^{10} - 727 p^{4} T^{11} + 234 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 11 T + 365 T^{2} + 3485 T^{3} + 63349 T^{4} + 508137 T^{5} + 98363 p T^{6} + 43382062 T^{7} + 98363 p^{2} T^{8} + 508137 p^{2} T^{9} + 63349 p^{3} T^{10} + 3485 p^{4} T^{11} + 365 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 19 T + 488 T^{2} - 6939 T^{3} + 100966 T^{4} - 1127717 T^{5} + 11738331 T^{6} - 103702974 T^{7} + 11738331 p T^{8} - 1127717 p^{2} T^{9} + 100966 p^{3} T^{10} - 6939 p^{4} T^{11} + 488 p^{5} T^{12} - 19 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 4 T + 244 T^{2} - 911 T^{3} + 26990 T^{4} - 89232 T^{5} + 2046293 T^{6} - 6333402 T^{7} + 2046293 p T^{8} - 89232 p^{2} T^{9} + 26990 p^{3} T^{10} - 911 p^{4} T^{11} + 244 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 3 T + 295 T^{2} - 167 T^{3} + 45187 T^{4} + 6323 T^{5} + 5052605 T^{6} + 138938 T^{7} + 5052605 p T^{8} + 6323 p^{2} T^{9} + 45187 p^{3} T^{10} - 167 p^{4} T^{11} + 295 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 10 T + 280 T^{2} - 2049 T^{3} + 43082 T^{4} - 287058 T^{5} + 4617147 T^{6} - 26111914 T^{7} + 4617147 p T^{8} - 287058 p^{2} T^{9} + 43082 p^{3} T^{10} - 2049 p^{4} T^{11} + 280 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 14 T + 286 T^{2} - 3261 T^{3} + 48216 T^{4} - 452558 T^{5} + 5426465 T^{6} - 45750382 T^{7} + 5426465 p T^{8} - 452558 p^{2} T^{9} + 48216 p^{3} T^{10} - 3261 p^{4} T^{11} + 286 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 20 T + 364 T^{2} - 5351 T^{3} + 62582 T^{4} - 665184 T^{5} + 6656701 T^{6} - 64161290 T^{7} + 6656701 p T^{8} - 665184 p^{2} T^{9} + 62582 p^{3} T^{10} - 5351 p^{4} T^{11} + 364 p^{5} T^{12} - 20 p^{6} T^{13} + p^{7} T^{14} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.58543447629565057652150485845, −5.46223915646741045442955508211, −5.41291761448808785156999179362, −5.30418828524936826542057997110, −4.89967057827934202834567284470, −4.83378054281481143613877835864, −4.81036297525795151632538759382, −4.52283232555523052785202443890, −4.37666961015726822887622678359, −4.29966207828478236922049935424, −4.20401776745629819824722216068, −3.94859445295030938040547555186, −3.56271300277036219783181313410, −3.36128503489860265687777217154, −3.31934634327448903012219211154, −3.29145097039353148053848094244, −2.67092330210770374711399662941, −2.58710991672385597854912351617, −2.49039153366388295774766359304, −2.17445047679954689485036383225, −1.48771143052583669179360706904, −1.44349276988779600396311579928, −1.09485169881951146353264858793, −0.826262619246607098708726109515, −0.68424067416347120305067372947, 0.68424067416347120305067372947, 0.826262619246607098708726109515, 1.09485169881951146353264858793, 1.44349276988779600396311579928, 1.48771143052583669179360706904, 2.17445047679954689485036383225, 2.49039153366388295774766359304, 2.58710991672385597854912351617, 2.67092330210770374711399662941, 3.29145097039353148053848094244, 3.31934634327448903012219211154, 3.36128503489860265687777217154, 3.56271300277036219783181313410, 3.94859445295030938040547555186, 4.20401776745629819824722216068, 4.29966207828478236922049935424, 4.37666961015726822887622678359, 4.52283232555523052785202443890, 4.81036297525795151632538759382, 4.83378054281481143613877835864, 4.89967057827934202834567284470, 5.30418828524936826542057997110, 5.41291761448808785156999179362, 5.46223915646741045442955508211, 5.58543447629565057652150485845
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.