Dirichlet series
| L(s) = 1 | − 3-s − 4-s − 3·5-s − 5·7-s − 2·8-s − 9-s + 11-s + 12-s − 8·13-s + 3·15-s + 16-s − 3·17-s − 3·19-s + 3·20-s + 5·21-s + 7·23-s + 2·24-s − 25-s + 5·28-s − 3·29-s − 31-s + 4·32-s − 33-s + 15·35-s + 36-s − 6·37-s + 8·39-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s − 1.34·5-s − 1.88·7-s − 0.707·8-s − 1/3·9-s + 0.301·11-s + 0.288·12-s − 2.21·13-s + 0.774·15-s + 1/4·16-s − 0.727·17-s − 0.688·19-s + 0.670·20-s + 1.09·21-s + 1.45·23-s + 0.408·24-s − 1/5·25-s + 0.944·28-s − 0.557·29-s − 0.179·31-s + 0.707·32-s − 0.174·33-s + 2.53·35-s + 1/6·36-s − 0.986·37-s + 1.28·39-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 62914 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62914 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(62914\) = \(2 \cdot 83 \cdot 379\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.01145\) |
| Root analytic conductor: | \(1.41522\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 62914,\ (\ :1/2, 1/2),\ 1)\) |
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_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) | |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) | ||
| 379 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) | ||
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.3.b_c |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.5.d_k | |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.7.f_s | |
| 11 | $D_{4}$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.11.ab_ae | |
| 13 | $D_{4}$ | \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.13.i_bi | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.17.d_bi | |
| 19 | $D_{4}$ | \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.19.d_y | |
| 23 | $D_{4}$ | \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.23.ah_cc | |
| 29 | $D_{4}$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.29.d_i | |
| 31 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.31.b_k | |
| 37 | $D_{4}$ | \( 1 + 6 T + 51 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.37.g_bz | |
| 41 | $D_{4}$ | \( 1 - T + 8 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.41.ab_i | |
| 43 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.43.ac_k | |
| 47 | $D_{4}$ | \( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.47.ae_ba | |
| 53 | $D_{4}$ | \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.53.c_bu | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.59.ad_am | |
| 61 | $D_{4}$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.61.g_bj | |
| 67 | $D_{4}$ | \( 1 - 5 T + 74 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.67.af_cw | |
| 71 | $D_{4}$ | \( 1 - 6 T + 106 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.71.ag_ec | |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.73.a_bu | |
| 79 | $D_{4}$ | \( 1 + 7 T + 116 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.79.h_em | |
| 89 | $D_{4}$ | \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.89.i_dm | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.97.m_gk | |
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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
−15.1340438324, −14.6098595397, −14.1725406418, −13.4106810622, −13.0503962123, −12.5315392824, −12.2998668390, −12.0096148758, −11.2969637532, −11.1183849743, −10.2933559266, −9.81136310892, −9.43182898044, −9.03749427712, −8.52360152707, −7.75505798253, −7.29930511271, −6.71070271070, −6.45853076820, −5.57567076702, −5.15339853113, −4.31306374758, −3.84320472528, −3.09386413008, −2.54037895281, 0, 0, 2.54037895281, 3.09386413008, 3.84320472528, 4.31306374758, 5.15339853113, 5.57567076702, 6.45853076820, 6.71070271070, 7.29930511271, 7.75505798253, 8.52360152707, 9.03749427712, 9.43182898044, 9.81136310892, 10.2933559266, 11.1183849743, 11.2969637532, 12.0096148758, 12.2998668390, 12.5315392824, 13.0503962123, 13.4106810622, 14.1725406418, 14.6098595397, 15.1340438324