Dirichlet series
| L(s) = 1 | − 3-s − 2·4-s − 2·5-s − 2·7-s + 9-s − 3·11-s + 2·12-s − 4·13-s + 2·15-s + 4·16-s + 17-s − 7·19-s + 4·20-s + 2·21-s − 25-s − 4·27-s + 4·28-s + 4·29-s + 8·31-s + 3·33-s + 4·35-s − 2·36-s + 4·37-s + 4·39-s − 41-s − 2·43-s + 6·44-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.894·5-s − 0.755·7-s + 1/3·9-s − 0.904·11-s + 0.577·12-s − 1.10·13-s + 0.516·15-s + 16-s + 0.242·17-s − 1.60·19-s + 0.894·20-s + 0.436·21-s − 1/5·25-s − 0.769·27-s + 0.755·28-s + 0.742·29-s + 1.43·31-s + 0.522·33-s + 0.676·35-s − 1/3·36-s + 0.657·37-s + 0.640·39-s − 0.156·41-s − 0.304·43-s + 0.904·44-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(6400\) = \(2^{8} \cdot 5^{2}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.408069\) |
| Root analytic conductor: | \(0.799251\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 6400,\ (\ :1/2, 1/2),\ -1)\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ | |
|---|---|---|---|---|
| bad | 2 | $C_2$ | \( 1 + p T^{2} \) | |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) | ||
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T + p T^{2} ) \) | 2.3.b_a |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.7.c_c | |
| 11 | $D_{4}$ | \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.11.d_q | |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) | 2.13.e_be | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.17.ab_bc | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.19.h_bs | |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.23.a_k | |
| 29 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.29.ae_w | |
| 31 | $D_{4}$ | \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.31.ai_by | |
| 37 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.37.ae_bm | |
| 41 | $D_{4}$ | \( 1 + T + 4 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.41.b_e | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.43.c_abi | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.47.g_dq | |
| 53 | $D_{4}$ | \( 1 + 2 T - 38 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.53.c_abm | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.59.g_bu | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.61.ai_bm | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.67.ab_acy | |
| 71 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.71.ag_w | |
| 73 | $D_{4}$ | \( 1 + 5 T + 56 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.73.f_ce | |
| 79 | $D_{4}$ | \( 1 - 2 T + 146 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.79.ac_fq | |
| 83 | $D_{4}$ | \( 1 + 3 T + 76 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.83.d_cy | |
| 89 | $D_{4}$ | \( 1 - T + 76 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.89.ab_cy | |
| 97 | $D_{4}$ | \( 1 - 4 T + 86 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.97.ae_di | |
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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
−17.3131008331, −17.1036324399, −16.4337953184, −15.9040248484, −15.4808331744, −14.8895353592, −14.5548680458, −13.6707732980, −13.2220529296, −12.7983409282, −12.2201819177, −11.9390372820, −11.0837292473, −10.4957358592, −9.84723348465, −9.67376148874, −8.61576094027, −8.11715945046, −7.62621960272, −6.73707427024, −6.08319164291, −5.16696703951, −4.56027875131, −3.89091353295, −2.74724723688, 0, 2.74724723688, 3.89091353295, 4.56027875131, 5.16696703951, 6.08319164291, 6.73707427024, 7.62621960272, 8.11715945046, 8.61576094027, 9.67376148874, 9.84723348465, 10.4957358592, 11.0837292473, 11.9390372820, 12.2201819177, 12.7983409282, 13.2220529296, 13.6707732980, 14.5548680458, 14.8895353592, 15.4808331744, 15.9040248484, 16.4337953184, 17.1036324399, 17.3131008331