Dirichlet series
| L(s) = 1 | − 2-s − 4-s − 7-s + 8-s − 4·9-s − 3·11-s + 13-s + 14-s + 3·16-s − 5·17-s + 4·18-s + 3·22-s − 4·23-s − 4·25-s − 26-s + 28-s + 5·29-s − 6·31-s − 3·32-s + 5·34-s + 4·36-s − 7·37-s + 2·41-s + 2·43-s + 3·44-s + 4·46-s + 4·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.377·7-s + 0.353·8-s − 4/3·9-s − 0.904·11-s + 0.277·13-s + 0.267·14-s + 3/4·16-s − 1.21·17-s + 0.942·18-s + 0.639·22-s − 0.834·23-s − 4/5·25-s − 0.196·26-s + 0.188·28-s + 0.928·29-s − 1.07·31-s − 0.530·32-s + 0.857·34-s + 2/3·36-s − 1.15·37-s + 0.312·41-s + 0.304·43-s + 0.452·44-s + 0.589·46-s + 0.583·47-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 10810 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10810 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(10810\) = \(2 \cdot 5 \cdot 23 \cdot 47\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.689254\) |
| Root analytic conductor: | \(0.911160\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 10810,\ (\ :1/2, 1/2),\ -1)\) |
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$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) | ||
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 5 T + p T^{2} ) \) | ||
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 3 T + p T^{2} ) \) | ||
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) | 2.3.a_e |
| 7 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.7.b_k | |
| 11 | $D_{4}$ | \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.11.d_q | |
| 13 | $D_{4}$ | \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.13.ab_j | |
| 17 | $D_{4}$ | \( 1 + 5 T + 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.17.f_q | |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) | 2.19.a_k | |
| 29 | $D_{4}$ | \( 1 - 5 T + 21 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.29.af_v | |
| 31 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.31.g_bm | |
| 37 | $D_{4}$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.37.h_bg | |
| 41 | $D_{4}$ | \( 1 - 2 T - 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.41.ac_abu | |
| 43 | $D_{4}$ | \( 1 - 2 T - 2 p T^{3} + p^{2} T^{4} \) | 2.43.ac_a | |
| 53 | $D_{4}$ | \( 1 - 17 T + 152 T^{2} - 17 p T^{3} + p^{2} T^{4} \) | 2.53.ar_fw | |
| 59 | $D_{4}$ | \( 1 + 7 T + 78 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.59.h_da | |
| 61 | $D_{4}$ | \( 1 + 3 T - 48 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.61.d_abw | |
| 67 | $D_{4}$ | \( 1 - 2 T - 40 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.67.ac_abo | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) | 2.71.ak_fm | |
| 73 | $D_{4}$ | \( 1 + 8 T + 144 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.73.i_fo | |
| 79 | $D_{4}$ | \( 1 - 7 T + 148 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.79.ah_fs | |
| 83 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) | 2.83.a_adq | |
| 89 | $C_2^2$ | \( 1 + 124 T^{2} + p^{2} T^{4} \) | 2.89.a_eu | |
| 97 | $D_{4}$ | \( 1 + 13 T + 170 T^{2} + 13 p T^{3} + p^{2} T^{4} \) | 2.97.n_go | |
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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
−16.7462178739, −16.3951316623, −15.7731142815, −15.3939192561, −14.8127657144, −14.1934258427, −13.6878071505, −13.4815581320, −12.6974750401, −12.2558736874, −11.6619966190, −10.9703349783, −10.6219465999, −9.96372154499, −9.43877988951, −8.77352143799, −8.52168324617, −7.95391891458, −7.24802402107, −6.35079738090, −5.74352917145, −5.18844014508, −4.17406136087, −3.29464269561, −2.29710594033, 0, 2.29710594033, 3.29464269561, 4.17406136087, 5.18844014508, 5.74352917145, 6.35079738090, 7.24802402107, 7.95391891458, 8.52168324617, 8.77352143799, 9.43877988951, 9.96372154499, 10.6219465999, 10.9703349783, 11.6619966190, 12.2558736874, 12.6974750401, 13.4815581320, 13.6878071505, 14.1934258427, 14.8127657144, 15.3939192561, 15.7731142815, 16.3951316623, 16.7462178739