Dirichlet series
| L(s) = 1 | + 5.24e5·2-s − 7.35e8·3-s + 2.74e11·4-s − 1.62e13·5-s − 3.85e14·6-s + 1.60e16·7-s + 1.44e17·8-s − 3.51e18·9-s − 8.50e18·10-s − 1.67e20·11-s − 2.02e20·12-s − 1.32e21·13-s + 8.41e21·14-s + 1.19e22·15-s + 7.55e22·16-s − 4.96e23·17-s − 1.84e24·18-s − 1.14e25·19-s − 4.46e24·20-s − 1.18e25·21-s − 8.77e25·22-s − 6.66e26·23-s − 1.05e26·24-s − 1.55e27·25-s − 6.93e26·26-s + 5.56e27·27-s + 4.41e27·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.365·3-s + 1/2·4-s − 0.380·5-s − 0.258·6-s + 0.532·7-s + 0.353·8-s − 0.866·9-s − 0.269·10-s − 0.825·11-s − 0.182·12-s − 0.251·13-s + 0.376·14-s + 0.138·15-s + 1/4·16-s − 0.503·17-s − 0.612·18-s − 1.33·19-s − 0.190·20-s − 0.194·21-s − 0.583·22-s − 1.86·23-s − 0.129·24-s − 0.855·25-s − 0.177·26-s + 0.681·27-s + 0.266·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(40-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+39/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(2\) |
| Sign: | $-1$ |
| Analytic conductor: | \(19.2679\) |
| Root analytic conductor: | \(4.38952\) |
| Motivic weight: | \(39\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((2,\ 2,\ (\ :39/2),\ -1)\) |
Particular Values
| \(L(20)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{41}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - p^{19} T \) |
| good | 3 | \( 1 + 9079732 p^{4} T + p^{39} T^{2} \) |
| 5 | \( 1 + 129809431866 p^{3} T + p^{39} T^{2} \) | |
| 7 | \( 1 - 2292866539412552 p T + p^{39} T^{2} \) | |
| 11 | \( 1 + 15220886148183602868 p T + p^{39} T^{2} \) | |
| 13 | \( 1 + \)\(10\!\cdots\!34\)\( p T + p^{39} T^{2} \) | |
| 17 | \( 1 + \)\(17\!\cdots\!94\)\( p^{2} T + p^{39} T^{2} \) | |
| 19 | \( 1 + \)\(60\!\cdots\!20\)\( p T + p^{39} T^{2} \) | |
| 23 | \( 1 + \)\(66\!\cdots\!72\)\( T + p^{39} T^{2} \) | |
| 29 | \( 1 - \)\(15\!\cdots\!50\)\( p T + p^{39} T^{2} \) | |
| 31 | \( 1 - \)\(15\!\cdots\!92\)\( T + p^{39} T^{2} \) | |
| 37 | \( 1 + \)\(69\!\cdots\!58\)\( p T + p^{39} T^{2} \) | |
| 41 | \( 1 - \)\(51\!\cdots\!42\)\( T + p^{39} T^{2} \) | |
| 43 | \( 1 - \)\(78\!\cdots\!28\)\( T + p^{39} T^{2} \) | |
| 47 | \( 1 - \)\(24\!\cdots\!04\)\( T + p^{39} T^{2} \) | |
| 53 | \( 1 - \)\(69\!\cdots\!98\)\( T + p^{39} T^{2} \) | |
| 59 | \( 1 + \)\(20\!\cdots\!00\)\( T + p^{39} T^{2} \) | |
| 61 | \( 1 + \)\(12\!\cdots\!18\)\( T + p^{39} T^{2} \) | |
| 67 | \( 1 - \)\(45\!\cdots\!64\)\( T + p^{39} T^{2} \) | |
| 71 | \( 1 + \)\(99\!\cdots\!28\)\( T + p^{39} T^{2} \) | |
| 73 | \( 1 - \)\(81\!\cdots\!18\)\( T + p^{39} T^{2} \) | |
| 79 | \( 1 + \)\(85\!\cdots\!40\)\( T + p^{39} T^{2} \) | |
| 83 | \( 1 - \)\(71\!\cdots\!48\)\( T + p^{39} T^{2} \) | |
| 89 | \( 1 + \)\(13\!\cdots\!90\)\( T + p^{39} T^{2} \) | |
| 97 | \( 1 - \)\(76\!\cdots\!34\)\( T + p^{39} T^{2} \) | |
| 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
−17.58342460668778859415805672800, −15.76001905867721474706666976285, −14.12477162382757872031745634333, −12.21831936355779020696548705989, −10.80569525993874776600701524501, −8.063512381102993696726985795027, −5.98118294901305269554181450553, −4.39472302074607937542041430925, −2.38168870150251759706220308002, 0, 2.38168870150251759706220308002, 4.39472302074607937542041430925, 5.98118294901305269554181450553, 8.063512381102993696726985795027, 10.80569525993874776600701524501, 12.21831936355779020696548705989, 14.12477162382757872031745634333, 15.76001905867721474706666976285, 17.58342460668778859415805672800