Dirichlet series
| L(s) = 1 | − 2-s − 2·3-s + 4-s − 2·5-s + 2·6-s − 4·7-s − 3·8-s + 2·10-s − 3·11-s − 2·12-s − 3·13-s + 4·14-s + 4·15-s + 16-s − 3·17-s − 6·19-s − 2·20-s + 8·21-s + 3·22-s − 23-s + 6·24-s − 2·25-s + 3·26-s + 2·27-s − 4·28-s + 2·29-s − 4·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s − 1.51·7-s − 1.06·8-s + 0.632·10-s − 0.904·11-s − 0.577·12-s − 0.832·13-s + 1.06·14-s + 1.03·15-s + 1/4·16-s − 0.727·17-s − 1.37·19-s − 0.447·20-s + 1.74·21-s + 0.639·22-s − 0.208·23-s + 1.22·24-s − 2/5·25-s + 0.588·26-s + 0.384·27-s − 0.755·28-s + 0.371·29-s − 0.730·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 45277 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45277 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(45277\) = \(19 \cdot 2383\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.88690\) |
| Root analytic conductor: | \(1.30349\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 45277,\ (\ :1/2, 1/2),\ 1)\) |
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 | 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 5 T + p T^{2} ) \) | |
| 2383 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 31 T + p T^{2} ) \) | ||
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) | 2.2.b_a |
| 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.3.c_e | |
| 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.5.c_g | |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.7.e_o | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.11.d_e | |
| 13 | $D_{4}$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.13.d_e | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.17.d_bi | |
| 23 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.23.b_ae | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.29.ac_bi | |
| 31 | $D_{4}$ | \( 1 - 7 T + 19 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.31.ah_t | |
| 37 | $D_{4}$ | \( 1 + 11 T + 83 T^{2} + 11 p T^{3} + p^{2} T^{4} \) | 2.37.l_df | |
| 41 | $D_{4}$ | \( 1 + 2 T - 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.41.c_ai | |
| 43 | $D_{4}$ | \( 1 - 14 T + 132 T^{2} - 14 p T^{3} + p^{2} T^{4} \) | 2.43.ao_fc | |
| 47 | $D_{4}$ | \( 1 - 9 T + 82 T^{2} - 9 p T^{3} + p^{2} T^{4} \) | 2.47.aj_de | |
| 53 | $D_{4}$ | \( 1 + 3 T + 79 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.53.d_db | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.59.ah_cw | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.61.c_cw | |
| 67 | $D_{4}$ | \( 1 + 3 T - 41 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.67.d_abp | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) | 2.71.t_hu | |
| 73 | $D_{4}$ | \( 1 - 14 T + 164 T^{2} - 14 p T^{3} + p^{2} T^{4} \) | 2.73.ao_gi | |
| 79 | $D_{4}$ | \( 1 + 11 T + 142 T^{2} + 11 p T^{3} + p^{2} T^{4} \) | 2.79.l_fm | |
| 83 | $D_{4}$ | \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \) | 2.83.q_fi | |
| 89 | $D_{4}$ | \( 1 + 24 T + 280 T^{2} + 24 p T^{3} + p^{2} T^{4} \) | 2.89.y_ku | |
| 97 | $D_{4}$ | \( 1 + 5 T - 63 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.97.f_acl | |
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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
−15.4732083519, −15.3060709660, −14.4819646030, −13.9243781669, −13.3112837764, −12.7687842453, −12.3567124030, −12.1271800568, −11.5599680536, −11.2014997430, −10.5985774802, −10.2612180293, −9.76746220267, −9.16720634397, −8.61220491376, −8.25953003805, −7.42900183744, −6.95042060148, −6.48410064877, −5.94454935157, −5.54271145439, −4.57061523313, −3.98768855770, −2.90916044875, −2.48540363876, 0, 0, 2.48540363876, 2.90916044875, 3.98768855770, 4.57061523313, 5.54271145439, 5.94454935157, 6.48410064877, 6.95042060148, 7.42900183744, 8.25953003805, 8.61220491376, 9.16720634397, 9.76746220267, 10.2612180293, 10.5985774802, 11.2014997430, 11.5599680536, 12.1271800568, 12.3567124030, 12.7687842453, 13.3112837764, 13.9243781669, 14.4819646030, 15.3060709660, 15.4732083519