Dirichlet series
| L(s) = 1 | + 7.35e8·3-s − 1.62e13·5-s − 1.60e16·7-s − 3.51e18·9-s + 1.67e20·11-s − 1.32e21·13-s − 1.19e22·15-s − 4.96e23·17-s + 1.14e25·19-s − 1.18e25·21-s + 6.66e26·23-s − 1.55e27·25-s − 5.56e27·27-s + 4.40e28·29-s − 1.58e28·31-s + 1.23e29·33-s + 2.60e29·35-s − 2.58e30·37-s − 9.73e29·39-s + 5.12e31·41-s − 7.87e31·43-s + 5.69e31·45-s − 2.42e32·47-s − 6.51e32·49-s − 3.65e32·51-s + 6.95e32·53-s − 2.71e33·55-s + ⋯ |
| L(s) = 1 | + 0.365·3-s − 0.380·5-s − 0.532·7-s − 0.866·9-s + 0.825·11-s − 0.251·13-s − 0.138·15-s − 0.503·17-s + 1.33·19-s − 0.194·21-s + 1.86·23-s − 0.855·25-s − 0.681·27-s + 1.33·29-s − 0.131·31-s + 0.301·33-s + 0.202·35-s − 0.680·37-s − 0.0917·39-s + 1.82·41-s − 1.10·43-s + 0.329·45-s − 0.601·47-s − 0.716·49-s − 0.184·51-s + 0.165·53-s − 0.314·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(40-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+39/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(16\) = \(2^{4}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(154.143\) |
| Root analytic conductor: | \(12.4154\) |
| Motivic weight: | \(39\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((2,\ 16,\ (\ :39/2),\ -1)\) |
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 | \( 1 \) |
| 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
−11.16040470963478559453206667932, −9.589863253685115723577068485137, −8.728105750701548500358788788779, −7.43165924021111040496408280391, −6.28166314172973168179013675768, −4.91781959356521908054196069256, −3.51775402338461573049257708860, −2.73837826988522197726229375854, −1.19190143439043861699268187834, 0, 1.19190143439043861699268187834, 2.73837826988522197726229375854, 3.51775402338461573049257708860, 4.91781959356521908054196069256, 6.28166314172973168179013675768, 7.43165924021111040496408280391, 8.728105750701548500358788788779, 9.589863253685115723577068485137, 11.16040470963478559453206667932