Dirichlet series
| L(s) = 1 | + 48·2-s + 1.95e5·3-s − 3.35e7·4-s + 9.39e6·6-s − 3.90e10·7-s − 3.22e9·8-s − 8.08e11·9-s + 8.41e12·11-s − 6.56e12·12-s + 8.16e13·13-s − 1.87e12·14-s + 1.12e15·16-s + 2.51e15·17-s − 3.88e13·18-s − 6.08e15·19-s − 7.65e15·21-s + 4.04e14·22-s + 9.49e16·23-s − 6.30e14·24-s + 3.91e15·26-s − 3.24e17·27-s + 1.31e18·28-s − 2.71e17·29-s + 4.29e18·31-s + 1.62e17·32-s + 1.64e18·33-s + 1.20e17·34-s + ⋯ |
| L(s) = 1 | + 0.00828·2-s + 0.212·3-s − 0.999·4-s + 0.00176·6-s − 1.06·7-s − 0.0165·8-s − 0.954·9-s + 0.808·11-s − 0.212·12-s + 0.972·13-s − 0.00884·14-s + 0.999·16-s + 1.04·17-s − 0.00791·18-s − 0.630·19-s − 0.227·21-s + 0.00670·22-s + 0.903·23-s − 0.00352·24-s + 0.00805·26-s − 0.415·27-s + 1.06·28-s − 0.142·29-s + 0.978·31-s + 0.0248·32-s + 0.172·33-s + 0.00869·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(25\) = \(5^{2}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(98.9991\) |
| Root analytic conductor: | \(9.94983\) |
| Motivic weight: | \(25\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((2,\ 25,\ (\ :25/2),\ -1)\) |
Particular Values
| \(L(13)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{27}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 3 p^{4} T + p^{25} T^{2} \) |
| 3 | \( 1 - 7252 p^{3} T + p^{25} T^{2} \) | |
| 7 | \( 1 + 797563208 p^{2} T + p^{25} T^{2} \) | |
| 11 | \( 1 - 765410481732 p T + p^{25} T^{2} \) | |
| 13 | \( 1 - 6280849641178 p T + p^{25} T^{2} \) | |
| 17 | \( 1 - 148229413467534 p T + p^{25} T^{2} \) | |
| 19 | \( 1 + 320108230016260 p T + p^{25} T^{2} \) | |
| 23 | \( 1 - 4130229578100888 p T + p^{25} T^{2} \) | |
| 29 | \( 1 + 271246959476737410 T + p^{25} T^{2} \) | |
| 31 | \( 1 - 4291666067521509152 T + p^{25} T^{2} \) | |
| 37 | \( 1 + 20301484446109126982 T + p^{25} T^{2} \) | |
| 41 | \( 1 + \)\(18\!\cdots\!98\)\( T + p^{25} T^{2} \) | |
| 43 | \( 1 + \)\(30\!\cdots\!56\)\( T + p^{25} T^{2} \) | |
| 47 | \( 1 - \)\(92\!\cdots\!88\)\( T + p^{25} T^{2} \) | |
| 53 | \( 1 - \)\(99\!\cdots\!54\)\( T + p^{25} T^{2} \) | |
| 59 | \( 1 - \)\(13\!\cdots\!80\)\( T + p^{25} T^{2} \) | |
| 61 | \( 1 - \)\(90\!\cdots\!02\)\( T + p^{25} T^{2} \) | |
| 67 | \( 1 - \)\(26\!\cdots\!28\)\( T + p^{25} T^{2} \) | |
| 71 | \( 1 + \)\(19\!\cdots\!48\)\( T + p^{25} T^{2} \) | |
| 73 | \( 1 + \)\(42\!\cdots\!26\)\( T + p^{25} T^{2} \) | |
| 79 | \( 1 + \)\(27\!\cdots\!60\)\( T + p^{25} T^{2} \) | |
| 83 | \( 1 - \)\(93\!\cdots\!84\)\( T + p^{25} T^{2} \) | |
| 89 | \( 1 + \)\(17\!\cdots\!30\)\( T + p^{25} T^{2} \) | |
| 97 | \( 1 + \)\(28\!\cdots\!62\)\( T + p^{25} T^{2} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.84917065700214288878612327665, −10.24122529044301190749911641186, −9.114714020103657442104761212448, −8.351111918090563906277154859850, −6.54868885652114982948526183948, −5.43709289590588417117475577835, −3.85771857050019556237294319586, −3.07126600820310473070498981041, −1.13438532337175879372469856421, 0, 1.13438532337175879372469856421, 3.07126600820310473070498981041, 3.85771857050019556237294319586, 5.43709289590588417117475577835, 6.54868885652114982948526183948, 8.351111918090563906277154859850, 9.114714020103657442104761212448, 10.24122529044301190749911641186, 11.84917065700214288878612327665