Dirichlet series
| L(s) = 1 | + 4·3-s − 2·5-s + 6·9-s + 4·11-s − 4·13-s − 8·15-s − 4·17-s − 4·19-s + 3·25-s − 4·27-s + 4·29-s + 16·33-s + 4·37-s − 16·39-s − 4·41-s + 12·43-s − 12·45-s + 8·47-s − 10·49-s − 16·51-s − 4·53-s − 8·55-s − 16·57-s − 12·59-s + 4·61-s + 8·65-s + 4·67-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 0.894·5-s + 2·9-s + 1.20·11-s − 1.10·13-s − 2.06·15-s − 0.970·17-s − 0.917·19-s + 3/5·25-s − 0.769·27-s + 0.742·29-s + 2.78·33-s + 0.657·37-s − 2.56·39-s − 0.624·41-s + 1.82·43-s − 1.78·45-s + 1.16·47-s − 1.42·49-s − 2.24·51-s − 0.549·53-s − 1.07·55-s − 2.11·57-s − 1.56·59-s + 0.512·61-s + 0.992·65-s + 0.488·67-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 12800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(12800\) = \(2^{9} \cdot 5^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.816139\) |
| Root analytic conductor: | \(0.950475\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 12800,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.712191947\) |
| \(L(\frac12)\) | \(\approx\) | \(1.712191947\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ | |
|---|---|---|---|---|
| bad | 2 | \( 1 \) | ||
| 5 | $C_1$ | \( ( 1 + T )^{2} \) | ||
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.3.ae_k |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.7.a_k | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) | 2.11.ae_w | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.13.e_o | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.17.e_w | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.19.e_g | |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.23.a_k | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.29.ae_bu | |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.31.a_bu | |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.37.ae_da | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.41.e_w | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.43.am_ec | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.47.ai_ec | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.53.e_dq | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.59.m_eo | |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.61.ae_ew | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.67.ae_es | |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) | 2.71.ay_la | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.73.am_gk | |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.79.a_dq | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.83.ae_ec | |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) | 2.89.m_ig | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.97.am_ig | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.9279585343, −15.6170085946, −15.1064991300, −14.8031968985, −14.2093345164, −14.2009317272, −13.5719193375, −12.9016643598, −12.4108590116, −11.9654378449, −11.1097308641, −10.9434418024, −9.68356529854, −9.57258761944, −8.74029332435, −8.73134517396, −7.85140754717, −7.71273110823, −6.81951269322, −6.27087624193, −4.92945525259, −4.12250433369, −3.69518610983, −2.76929890617, −2.17341817165, 2.17341817165, 2.76929890617, 3.69518610983, 4.12250433369, 4.92945525259, 6.27087624193, 6.81951269322, 7.71273110823, 7.85140754717, 8.73134517396, 8.74029332435, 9.57258761944, 9.68356529854, 10.9434418024, 11.1097308641, 11.9654378449, 12.4108590116, 12.9016643598, 13.5719193375, 14.2009317272, 14.2093345164, 14.8031968985, 15.1064991300, 15.6170085946, 15.9279585343