Dirichlet series
| L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s − 2·5-s + 4·6-s − 2·7-s + 9-s + 4·10-s − 6·11-s − 4·12-s − 13-s + 4·14-s + 4·15-s − 4·16-s − 12·17-s − 2·18-s − 19-s − 4·20-s + 4·21-s + 12·22-s + 3·23-s + 25-s + 2·26-s − 2·27-s − 4·28-s − 2·29-s − 8·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s − 0.894·5-s + 1.63·6-s − 0.755·7-s + 1/3·9-s + 1.26·10-s − 1.80·11-s − 1.15·12-s − 0.277·13-s + 1.06·14-s + 1.03·15-s − 16-s − 2.91·17-s − 0.471·18-s − 0.229·19-s − 0.894·20-s + 0.872·21-s + 2.55·22-s + 0.625·23-s + 1/5·25-s + 0.392·26-s − 0.384·27-s − 0.755·28-s − 0.371·29-s − 1.46·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 46432 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46432 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(46432\) = \(2^{5} \cdot 1451\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.96054\) |
| Root analytic conductor: | \(1.31172\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 46432,\ (\ :1/2, 1/2),\ 1)\) |
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 | $C_2$ | \( 1 + p T + p T^{2} \) | |
| 1451 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 65 T + p T^{2} ) \) | ||
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) | 2.3.c_d |
| 5 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.5.c_d | |
| 7 | $D_{4}$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.7.c_b | |
| 11 | $D_{4}$ | \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.11.g_u | |
| 13 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) | 2.13.b_n | |
| 17 | $D_{4}$ | \( 1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.17.m_cp | |
| 19 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.19.b_f | |
| 23 | $D_{4}$ | \( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.23.ad_bo | |
| 29 | $D_{4}$ | \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.29.c_t | |
| 31 | $D_{4}$ | \( 1 + 5 T + 52 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.31.f_ca | |
| 37 | $D_{4}$ | \( 1 - 9 T + 45 T^{2} - 9 p T^{3} + p^{2} T^{4} \) | 2.37.aj_bt | |
| 41 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) | 2.41.a_ach | |
| 43 | $D_{4}$ | \( 1 + 5 T + 60 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.43.f_ci | |
| 47 | $D_{4}$ | \( 1 + 13 T + 97 T^{2} + 13 p T^{3} + p^{2} T^{4} \) | 2.47.n_dt | |
| 53 | $D_{4}$ | \( 1 + 3 T - 59 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.53.d_ach | |
| 59 | $D_{4}$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.59.m_cz | |
| 61 | $D_{4}$ | \( 1 - 2 T - 24 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.61.ac_ay | |
| 67 | $D_{4}$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.67.ac_az | |
| 71 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) | 2.71.a_bs | |
| 73 | $D_{4}$ | \( 1 - 2 T + 136 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.73.ac_fg | |
| 79 | $D_{4}$ | \( 1 + 3 T - p T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.79.d_adb | |
| 83 | $D_{4}$ | \( 1 + T + 42 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.83.b_bq | |
| 89 | $D_{4}$ | \( 1 - 6 T + 44 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.89.ag_bs | |
| 97 | $D_{4}$ | \( 1 - 13 T + 149 T^{2} - 13 p T^{3} + p^{2} T^{4} \) | 2.97.an_ft | |
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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
−15.6596162689, −15.1054038288, −14.6958689257, −13.5569253872, −13.3177278659, −12.9015306040, −12.6428843610, −11.6221784411, −11.4124998302, −11.0950373748, −10.6330623046, −10.3399834962, −9.51634865586, −9.26299865727, −8.61822330992, −8.09272674080, −7.61557203811, −7.13139424818, −6.49734745909, −6.16955021682, −5.11591576590, −4.78661948708, −4.02903871537, −2.87292125788, −2.08359670066, 0, 0, 2.08359670066, 2.87292125788, 4.02903871537, 4.78661948708, 5.11591576590, 6.16955021682, 6.49734745909, 7.13139424818, 7.61557203811, 8.09272674080, 8.61822330992, 9.26299865727, 9.51634865586, 10.3399834962, 10.6330623046, 11.0950373748, 11.4124998302, 11.6221784411, 12.6428843610, 12.9015306040, 13.3177278659, 13.5569253872, 14.6958689257, 15.1054038288, 15.6596162689