Dirichlet series
| L(s) = 1 | − 3·9-s + 4·13-s + 2·17-s − 4·29-s − 12·37-s − 10·41-s − 7·49-s + 4·53-s + 12·61-s + 6·73-s + 9·81-s + 10·89-s + 18·97-s − 20·101-s − 20·109-s + 14·113-s − 12·117-s + ⋯ |
| L(s) = 1 | − 9-s + 1.10·13-s + 0.485·17-s − 0.742·29-s − 1.97·37-s − 1.56·41-s − 49-s + 0.549·53-s + 1.53·61-s + 0.702·73-s + 81-s + 1.05·89-s + 1.82·97-s − 1.99·101-s − 1.91·109-s + 1.31·113-s − 1.10·117-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(6400\) = \(2^{8} \cdot 5^{2}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(51.1042\) |
| Root analytic conductor: | \(7.14872\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((2,\ 6400,\ (\ :1/2),\ -1)\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ | |
|---|---|---|---|
| bad | 2 | \( 1 \) | |
| 5 | \( 1 \) | ||
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a | |
| 11 | \( 1 + p T^{2} \) | 1.11.a | |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae | |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac | |
| 19 | \( 1 + p T^{2} \) | 1.19.a | |
| 23 | \( 1 + p T^{2} \) | 1.23.a | |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e | |
| 31 | \( 1 + p T^{2} \) | 1.31.a | |
| 37 | \( 1 + 12 T + p T^{2} \) | 1.37.m | |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k | |
| 43 | \( 1 + p T^{2} \) | 1.43.a | |
| 47 | \( 1 + p T^{2} \) | 1.47.a | |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae | |
| 59 | \( 1 + p T^{2} \) | 1.59.a | |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am | |
| 67 | \( 1 + p T^{2} \) | 1.67.a | |
| 71 | \( 1 + p T^{2} \) | 1.71.a | |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag | |
| 79 | \( 1 + p T^{2} \) | 1.79.a | |
| 83 | \( 1 + p T^{2} \) | 1.83.a | |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak | |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.78467951471813866703142211403, −6.87315856118033060742083453653, −6.27764363430354085556091551464, −5.47237371652670835325377298000, −5.02456163691776154135027092680, −3.70688844144968490550979770674, −3.41512333131181533280573297684, −2.29881549788790531514484661276, −1.32110387882551793301287621394, 0, 1.32110387882551793301287621394, 2.29881549788790531514484661276, 3.41512333131181533280573297684, 3.70688844144968490550979770674, 5.02456163691776154135027092680, 5.47237371652670835325377298000, 6.27764363430354085556091551464, 6.87315856118033060742083453653, 7.78467951471813866703142211403