Dirichlet series
| L(s) = 1 | − 2-s − 3-s − 2·5-s + 6-s + 7-s − 8-s + 9-s + 2·10-s − 2·11-s + 2·13-s − 14-s + 2·15-s − 16-s − 3·17-s − 18-s − 2·19-s − 21-s + 2·22-s + 23-s + 24-s − 2·25-s − 2·26-s − 27-s − 2·30-s − 8·31-s + 6·32-s + 2·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 0.894·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.603·11-s + 0.554·13-s − 0.267·14-s + 0.516·15-s − 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.458·19-s − 0.218·21-s + 0.426·22-s + 0.208·23-s + 0.204·24-s − 2/5·25-s − 0.392·26-s − 0.192·27-s − 0.365·30-s − 1.43·31-s + 1.06·32-s + 0.348·33-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 12555 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12555 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(12555\) = \(3^{4} \cdot 5 \cdot 31\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.800517\) |
| Root analytic conductor: | \(0.945894\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 12555,\ (\ :1/2, 1/2),\ -1)\) |
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 | $C_1$ | \( 1 + T \) | |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) | ||
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 7 T + p T^{2} ) \) | ||
| good | 2 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) | 2.2.b_b |
| 7 | $D_{4}$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.7.ab_ac | |
| 11 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.11.c_ag | |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.13.ac_c | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.17.d_bi | |
| 19 | $D_{4}$ | \( 1 + 2 T - 17 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.19.c_ar | |
| 23 | $D_{4}$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.23.ab_ae | |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) | 2.29.a_ba | |
| 37 | $D_{4}$ | \( 1 + 10 T + 82 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.37.k_de | |
| 41 | $D_{4}$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.41.j_ca | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.43.k_ck | |
| 47 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.47.ai_ck | |
| 53 | $D_{4}$ | \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.53.h_be | |
| 59 | $D_{4}$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.59.ab_ak | |
| 61 | $D_{4}$ | \( 1 - T + 74 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.61.ab_cw | |
| 67 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.67.ac_ac | |
| 71 | $D_{4}$ | \( 1 - 3 T + 120 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.71.ad_eq | |
| 73 | $D_{4}$ | \( 1 - 14 T + 110 T^{2} - 14 p T^{3} + p^{2} T^{4} \) | 2.73.ao_eg | |
| 79 | $D_{4}$ | \( 1 - 21 T + 236 T^{2} - 21 p T^{3} + p^{2} T^{4} \) | 2.79.av_jc | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) | 2.83.b_abs | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.89.q_is | |
| 97 | $D_{4}$ | \( 1 + 7 T - 46 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.97.h_abu | |
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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
−16.6570097545, −16.0227303319, −15.6225158653, −15.1631327940, −15.0485187549, −13.9125172843, −13.6913001511, −13.0622455755, −12.3300071158, −12.1175822157, −11.3100349314, −11.1232613720, −10.5845880454, −9.96846073008, −9.26243060907, −8.74473265045, −8.24052584521, −7.79433689360, −6.85181852093, −6.64657372545, −5.54427020789, −5.02751138647, −4.13826527771, −3.41831178091, −1.99391543411, 0, 1.99391543411, 3.41831178091, 4.13826527771, 5.02751138647, 5.54427020789, 6.64657372545, 6.85181852093, 7.79433689360, 8.24052584521, 8.74473265045, 9.26243060907, 9.96846073008, 10.5845880454, 11.1232613720, 11.3100349314, 12.1175822157, 12.3300071158, 13.0622455755, 13.6913001511, 13.9125172843, 15.0485187549, 15.1631327940, 15.6225158653, 16.0227303319, 16.6570097545