Dirichlet series
| L(s) = 1 | − 3·2-s − 4·3-s + 4·4-s − 6·5-s + 12·6-s − 6·7-s − 3·8-s + 7·9-s + 18·10-s − 3·11-s − 16·12-s − 2·13-s + 18·14-s + 24·15-s + 3·16-s − 8·17-s − 21·18-s − 5·19-s − 24·20-s + 24·21-s + 9·22-s − 6·23-s + 12·24-s + 19·25-s + 6·26-s − 4·27-s − 24·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 2.30·3-s + 2·4-s − 2.68·5-s + 4.89·6-s − 2.26·7-s − 1.06·8-s + 7/3·9-s + 5.69·10-s − 0.904·11-s − 4.61·12-s − 0.554·13-s + 4.81·14-s + 6.19·15-s + 3/4·16-s − 1.94·17-s − 4.94·18-s − 1.14·19-s − 5.36·20-s + 5.23·21-s + 1.91·22-s − 1.25·23-s + 2.44·24-s + 19/5·25-s + 1.17·26-s − 0.769·27-s − 4.53·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 25913 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25913 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(25913\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.65223\) |
| Root analytic conductor: | \(1.13375\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(3\) |
| Selberg data: | \((4,\ 25913,\ (\ :1/2, 1/2),\ -1)\) |
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 | 25913 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 61 T + p T^{2} ) \) | |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.2.d_f |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) | 2.3.e_j | |
| 5 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.5.g_r | |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.7.g_w | |
| 11 | $D_{4}$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.11.d_c | |
| 13 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.13.c_ac | |
| 17 | $D_{4}$ | \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.17.i_bo | |
| 19 | $D_{4}$ | \( 1 + 5 T + 26 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.19.f_ba | |
| 23 | $D_{4}$ | \( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.23.g_bf | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.29.e_ac | |
| 31 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) | 2.31.a_al | |
| 37 | $D_{4}$ | \( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.37.ad_ca | |
| 41 | $D_{4}$ | \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.41.g_cw | |
| 43 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) | 2.43.a_bx | |
| 47 | $D_{4}$ | \( 1 + 2 T - 40 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.47.c_abo | |
| 53 | $D_{4}$ | \( 1 + T + 29 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.53.b_bd | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.59.ac_cs | |
| 61 | $C_2^2$ | \( 1 - 27 T^{2} + p^{2} T^{4} \) | 2.61.a_abb | |
| 67 | $D_{4}$ | \( 1 + 10 T + 137 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.67.k_fh | |
| 71 | $D_{4}$ | \( 1 + 6 T - 8 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.71.g_ai | |
| 73 | $D_{4}$ | \( 1 + 6 T + 4 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.73.g_e | |
| 79 | $D_{4}$ | \( 1 - 6 T + 48 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.79.ag_bw | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.83.c_gc | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.89.e_gb | |
| 97 | $D_{4}$ | \( 1 + 11 T + 98 T^{2} + 11 p T^{3} + p^{2} T^{4} \) | 2.97.l_du | |
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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
−16.2693591004, −16.0184147193, −15.7560995960, −15.1561219453, −14.9085332030, −13.3400755218, −13.0641035256, −12.4137825210, −12.0936918254, −11.8250306288, −11.1065877350, −10.8782225807, −10.5122563592, −9.92017296890, −9.41783196267, −8.75787920906, −8.21080790407, −7.80905325357, −7.05464350804, −6.70696518394, −6.24749066645, −5.45648770054, −4.42915910934, −3.97385895732, −2.89022073030, 0, 0, 0, 2.89022073030, 3.97385895732, 4.42915910934, 5.45648770054, 6.24749066645, 6.70696518394, 7.05464350804, 7.80905325357, 8.21080790407, 8.75787920906, 9.41783196267, 9.92017296890, 10.5122563592, 10.8782225807, 11.1065877350, 11.8250306288, 12.0936918254, 12.4137825210, 13.0641035256, 13.3400755218, 14.9085332030, 15.1561219453, 15.7560995960, 16.0184147193, 16.2693591004