Dirichlet series
| L(s) = 1 | − 2·2-s − 3-s + 4-s + 2·6-s − 3·7-s − 2·9-s − 7·11-s − 12-s − 2·13-s + 6·14-s + 16-s − 3·17-s + 4·18-s + 4·19-s + 3·21-s + 14·22-s − 11·23-s − 10·25-s + 4·26-s + 2·27-s − 3·28-s − 2·29-s + 3·31-s + 2·32-s + 7·33-s + 6·34-s − 2·36-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 1/2·4-s + 0.816·6-s − 1.13·7-s − 2/3·9-s − 2.11·11-s − 0.288·12-s − 0.554·13-s + 1.60·14-s + 1/4·16-s − 0.727·17-s + 0.942·18-s + 0.917·19-s + 0.654·21-s + 2.98·22-s − 2.29·23-s − 2·25-s + 0.784·26-s + 0.384·27-s − 0.566·28-s − 0.371·29-s + 0.538·31-s + 0.353·32-s + 1.21·33-s + 1.02·34-s − 1/3·36-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 53823 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53823 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(53823\) = \(3 \cdot 7 \cdot 11 \cdot 233\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.43180\) |
| Root analytic conductor: | \(1.36107\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 53823,\ (\ :1/2, 1/2),\ 1)\) |
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 | 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) | |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) | ||
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 6 T + p T^{2} ) \) | ||
| 233 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 26 T + p T^{2} ) \) | ||
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) | 2.2.c_d |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.5.a_k | |
| 13 | $C_4$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.13.c_k | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.17.d_bi | |
| 19 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.19.ae_w | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) | 2.23.l_cm | |
| 29 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.29.c_w | |
| 31 | $D_{4}$ | \( 1 - 3 T + 6 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.31.ad_g | |
| 37 | $D_{4}$ | \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.37.ad_ae | |
| 41 | $D_{4}$ | \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.41.f_be | |
| 43 | $D_{4}$ | \( 1 + 3 T - 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.43.d_ag | |
| 47 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.47.ac_k | |
| 53 | $D_{4}$ | \( 1 + 2 T + 94 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.53.c_dq | |
| 59 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.59.e_g | |
| 61 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) | 2.61.a_abq | |
| 67 | $D_{4}$ | \( 1 - 16 T + 158 T^{2} - 16 p T^{3} + p^{2} T^{4} \) | 2.67.aq_gc | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) | 2.71.n_ei | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) | 2.73.j_ce | |
| 79 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.79.ab_s | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.83.ae_cs | |
| 89 | $D_{4}$ | \( 1 + 9 T + 158 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.89.j_gc | |
| 97 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.97.c_bq | |
| show more | ||||
| show less | ||||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.3544649345, −14.9279834743, −13.9392756570, −13.7712579562, −13.3401777204, −12.8538674656, −12.2177545116, −11.8282251670, −11.5030907558, −10.7543324567, −10.2761542906, −9.91534115133, −9.66794580812, −9.21865739748, −8.36526624938, −8.11349259878, −7.72225363827, −7.11302578547, −6.23457101704, −5.84913554964, −5.44939876076, −4.61824301576, −3.71039626813, −2.83194850684, −2.17935114663, 0, 0, 2.17935114663, 2.83194850684, 3.71039626813, 4.61824301576, 5.44939876076, 5.84913554964, 6.23457101704, 7.11302578547, 7.72225363827, 8.11349259878, 8.36526624938, 9.21865739748, 9.66794580812, 9.91534115133, 10.2761542906, 10.7543324567, 11.5030907558, 11.8282251670, 12.2177545116, 12.8538674656, 13.3401777204, 13.7712579562, 13.9392756570, 14.9279834743, 15.3544649345