Dirichlet series
| L(s) = 1 | − 2-s − 2·3-s + 4-s − 4·5-s + 2·6-s − 2·7-s − 3·8-s − 2·9-s + 4·10-s − 11-s − 2·12-s − 7·13-s + 2·14-s + 8·15-s + 16-s − 3·17-s + 2·18-s − 4·20-s + 4·21-s + 22-s + 3·23-s + 6·24-s + 6·25-s + 7·26-s + 10·27-s − 2·28-s + 3·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 1.78·5-s + 0.816·6-s − 0.755·7-s − 1.06·8-s − 2/3·9-s + 1.26·10-s − 0.301·11-s − 0.577·12-s − 1.94·13-s + 0.534·14-s + 2.06·15-s + 1/4·16-s − 0.727·17-s + 0.471·18-s − 0.894·20-s + 0.872·21-s + 0.213·22-s + 0.625·23-s + 1.22·24-s + 6/5·25-s + 1.37·26-s + 1.92·27-s − 0.377·28-s + 0.557·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 31307 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31307 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(31307\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.99616\) |
| Root analytic conductor: | \(1.18863\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 31307,\ (\ :1/2, 1/2),\ 1)\) |
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 | 31307 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 108 T + p T^{2} ) \) | |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) | 2.2.b_a |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.3.c_g | |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.5.e_k | |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.7.c_g | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.11.b_k | |
| 13 | $D_{4}$ | \( 1 + 7 T + 29 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.13.h_bd | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.17.d_bi | |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) | 2.19.a_be | |
| 23 | $D_{4}$ | \( 1 - 3 T + 15 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.23.ad_p | |
| 29 | $D_{4}$ | \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.29.ad_m | |
| 31 | $D_{4}$ | \( 1 + 5 T + 19 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.31.f_t | |
| 37 | $D_{4}$ | \( 1 + T + 12 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.37.b_m | |
| 41 | $D_{4}$ | \( 1 - 4 T + 44 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.41.ae_bs | |
| 43 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.43.ac_bi | |
| 47 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) | 2.47.a_ca | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.53.f_ce | |
| 59 | $D_{4}$ | \( 1 + 14 T + 124 T^{2} + 14 p T^{3} + p^{2} T^{4} \) | 2.59.o_eu | |
| 61 | $D_{4}$ | \( 1 + 5 T + 88 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.61.f_dk | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.67.h_bk | |
| 71 | $D_{4}$ | \( 1 - 10 T + 102 T^{2} - 10 p T^{3} + p^{2} T^{4} \) | 2.71.ak_dy | |
| 73 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.73.ac_s | |
| 79 | $D_{4}$ | \( 1 + 21 T + 249 T^{2} + 21 p T^{3} + p^{2} T^{4} \) | 2.79.v_jp | |
| 83 | $D_{4}$ | \( 1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.83.d_z | |
| 89 | $D_{4}$ | \( 1 - 3 T + 175 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.89.ad_gt | |
| 97 | $D_{4}$ | \( 1 - T - 51 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.97.ab_abz | |
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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
−15.7132887929, −15.4925325973, −14.9900921398, −14.5013526684, −14.0445727790, −13.0886988216, −12.5784756333, −12.1377069096, −11.9807395115, −11.4444747481, −11.0484208752, −10.7340864456, −9.97594280109, −9.33968037334, −8.98955730637, −8.35303290441, −7.72128066915, −7.33594308656, −6.66867997620, −6.26376039414, −5.46494370296, −4.86931383236, −4.17833432975, −3.05601431083, −2.68951833635, 0, 0, 2.68951833635, 3.05601431083, 4.17833432975, 4.86931383236, 5.46494370296, 6.26376039414, 6.66867997620, 7.33594308656, 7.72128066915, 8.35303290441, 8.98955730637, 9.33968037334, 9.97594280109, 10.7340864456, 11.0484208752, 11.4444747481, 11.9807395115, 12.1377069096, 12.5784756333, 13.0886988216, 14.0445727790, 14.5013526684, 14.9900921398, 15.4925325973, 15.7132887929