Dirichlet series
| L(s) = 1 | − 3·2-s − 3·3-s + 4·4-s − 3·5-s + 9·6-s − 3·7-s − 3·8-s + 3·9-s + 9·10-s − 5·11-s − 12·12-s − 6·13-s + 9·14-s + 9·15-s + 3·16-s − 17-s − 9·18-s + 19-s − 12·20-s + 9·21-s + 15·22-s − 10·23-s + 9·24-s + 3·25-s + 18·26-s − 12·28-s − 27·30-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 2·4-s − 1.34·5-s + 3.67·6-s − 1.13·7-s − 1.06·8-s + 9-s + 2.84·10-s − 1.50·11-s − 3.46·12-s − 1.66·13-s + 2.40·14-s + 2.32·15-s + 3/4·16-s − 0.242·17-s − 2.12·18-s + 0.229·19-s − 2.68·20-s + 1.96·21-s + 3.19·22-s − 2.08·23-s + 1.83·24-s + 3/5·25-s + 3.53·26-s − 2.26·28-s − 4.92·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 213487 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213487 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(213487\) = \(41^{2} \cdot 127\) |
| Sign: | $-1$ |
| Analytic conductor: | \(13.6121\) |
| Root analytic conductor: | \(1.92079\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(3\) |
| Selberg data: | \((4,\ 213487,\ (\ :1/2, 1/2),\ -1)\) |
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 | 41 | $C_2$ | \( 1 - 2 T + p T^{2} \) | |
| 127 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 7 T + p T^{2} ) \) | ||
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.2.d_f |
| 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) | 2.3.d_g | |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.5.d_g | |
| 7 | $D_{4}$ | \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.7.d_g | |
| 11 | $D_{4}$ | \( 1 + 5 T + 21 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.11.f_v | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.13.g_ba | |
| 17 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) | 2.17.b_b | |
| 19 | $D_{4}$ | \( 1 - T + 15 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.19.ab_p | |
| 23 | $D_{4}$ | \( 1 + 10 T + 51 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.23.k_bz | |
| 29 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) | 2.29.a_ai | |
| 31 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.31.j_cg | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) | 2.37.g_t | |
| 43 | $D_{4}$ | \( 1 + 5 T + 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.43.f_bi | |
| 47 | $D_{4}$ | \( 1 + 3 T + T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.47.d_b | |
| 53 | $D_{4}$ | \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.53.c_at | |
| 59 | $D_{4}$ | \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \) | 2.59.q_fi | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.61.m_es | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.67.g_dq | |
| 71 | $D_{4}$ | \( 1 + 4 T - 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.71.e_aw | |
| 73 | $D_{4}$ | \( 1 - 3 T + 117 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.73.ad_en | |
| 79 | $D_{4}$ | \( 1 + 17 T + 207 T^{2} + 17 p T^{3} + p^{2} T^{4} \) | 2.79.r_hz | |
| 83 | $D_{4}$ | \( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.83.ad_q | |
| 89 | $D_{4}$ | \( 1 + 10 T + 133 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.89.k_fd | |
| 97 | $D_{4}$ | \( 1 - 6 T + 14 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.97.ag_o | |
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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
−13.9037123881, −13.4610990329, −12.6791948690, −12.4809163733, −12.1613431105, −11.7887236352, −11.3583209995, −10.8336609751, −10.4673192708, −10.2492227339, −9.70614994227, −9.43995592476, −8.89002587312, −8.22181225702, −7.88714611321, −7.51749366652, −7.20275430420, −6.58016157055, −5.81220157247, −5.69629018629, −4.91192526747, −4.39898185231, −3.47727531725, −2.85738683279, −1.79940037702, 0, 0, 0, 1.79940037702, 2.85738683279, 3.47727531725, 4.39898185231, 4.91192526747, 5.69629018629, 5.81220157247, 6.58016157055, 7.20275430420, 7.51749366652, 7.88714611321, 8.22181225702, 8.89002587312, 9.43995592476, 9.70614994227, 10.2492227339, 10.4673192708, 10.8336609751, 11.3583209995, 11.7887236352, 12.1613431105, 12.4809163733, 12.6791948690, 13.4610990329, 13.9037123881