Dirichlet series
| L(s) = 1 | − 2-s − 4-s + 2·5-s − 4·7-s + 8-s − 4·9-s − 2·10-s − 11-s − 7·13-s + 4·14-s + 3·16-s + 4·18-s − 6·19-s − 2·20-s + 22-s + 4·23-s + 2·25-s + 7·26-s + 4·28-s + 3·29-s − 3·31-s − 3·32-s − 8·35-s + 4·36-s + 3·37-s + 6·38-s + 2·40-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s − 1.51·7-s + 0.353·8-s − 4/3·9-s − 0.632·10-s − 0.301·11-s − 1.94·13-s + 1.06·14-s + 3/4·16-s + 0.942·18-s − 1.37·19-s − 0.447·20-s + 0.213·22-s + 0.834·23-s + 2/5·25-s + 1.37·26-s + 0.755·28-s + 0.557·29-s − 0.538·31-s − 0.530·32-s − 1.35·35-s + 2/3·36-s + 0.493·37-s + 0.973·38-s + 0.316·40-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 10130 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10130 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(10130\) = \(2 \cdot 5 \cdot 1013\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.645897\) |
| Root analytic conductor: | \(0.896480\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 10130,\ (\ :1/2, 1/2),\ -1)\) |
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$ | \( ( 1 + T )( 1 + p T^{2} ) \) | |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) | ||
| 1013 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 35 T + p T^{2} ) \) | ||
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) | 2.3.a_e |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.7.e_o | |
| 11 | $D_{4}$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.11.b_ac | |
| 13 | $D_{4}$ | \( 1 + 7 T + 33 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.13.h_bh | |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) | 2.17.a_w | |
| 19 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.19.g_bc | |
| 23 | $D_{4}$ | \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.23.ae_q | |
| 29 | $D_{4}$ | \( 1 - 3 T + 3 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.29.ad_d | |
| 31 | $D_{4}$ | \( 1 + 3 T + 54 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.31.d_cc | |
| 37 | $D_{4}$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.37.ad_abg | |
| 41 | $D_{4}$ | \( 1 - 6 T + 8 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.41.ag_i | |
| 43 | $D_{4}$ | \( 1 - 5 T + 11 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.43.af_l | |
| 47 | $D_{4}$ | \( 1 - 13 T + 121 T^{2} - 13 p T^{3} + p^{2} T^{4} \) | 2.47.an_er | |
| 53 | $D_{4}$ | \( 1 + 7 T + 100 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.53.h_dw | |
| 59 | $D_{4}$ | \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.59.d_j | |
| 61 | $D_{4}$ | \( 1 + 11 T + 67 T^{2} + 11 p T^{3} + p^{2} T^{4} \) | 2.61.l_cp | |
| 67 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) | 2.67.a_aby | |
| 71 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.71.e_cc | |
| 73 | $D_{4}$ | \( 1 + 12 T + 112 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.73.m_ei | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) | 2.79.d_aw | |
| 83 | $D_{4}$ | \( 1 - 7 T - 6 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.83.ah_ag | |
| 89 | $D_{4}$ | \( 1 + T - 99 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.89.b_adv | |
| 97 | $D_{4}$ | \( 1 + 20 T + 240 T^{2} + 20 p T^{3} + p^{2} T^{4} \) | 2.97.u_jg | |
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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
−16.9705009657, −16.6823097671, −16.0145032545, −15.2492916429, −14.8390964063, −14.2293951292, −14.0660916490, −13.1126867682, −12.9189322405, −12.4021606525, −11.9419072691, −10.9039897235, −10.5272409215, −9.96395120285, −9.38987967442, −9.17632518264, −8.67046031761, −7.81724768831, −7.22763325435, −6.31659580982, −5.92738277418, −5.21223927154, −4.34714388484, −2.99824975312, −2.54695632700, 0, 2.54695632700, 2.99824975312, 4.34714388484, 5.21223927154, 5.92738277418, 6.31659580982, 7.22763325435, 7.81724768831, 8.67046031761, 9.17632518264, 9.38987967442, 9.96395120285, 10.5272409215, 10.9039897235, 11.9419072691, 12.4021606525, 12.9189322405, 13.1126867682, 14.0660916490, 14.2293951292, 14.8390964063, 15.2492916429, 16.0145032545, 16.6823097671, 16.9705009657