Dirichlet series
| L(s) = 1 | − 2-s − 2·3-s − 4-s − 4·5-s + 2·6-s − 3·7-s + 8-s + 4·10-s − 11-s + 2·12-s − 3·13-s + 3·14-s + 8·15-s − 16-s − 3·17-s − 3·19-s + 4·20-s + 6·21-s + 22-s − 5·23-s − 2·24-s + 6·25-s + 3·26-s + 5·27-s + 3·28-s − 8·30-s − 31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s − 1.78·5-s + 0.816·6-s − 1.13·7-s + 0.353·8-s + 1.26·10-s − 0.301·11-s + 0.577·12-s − 0.832·13-s + 0.801·14-s + 2.06·15-s − 1/4·16-s − 0.727·17-s − 0.688·19-s + 0.894·20-s + 1.30·21-s + 0.213·22-s − 1.04·23-s − 0.408·24-s + 6/5·25-s + 0.588·26-s + 0.962·27-s + 0.566·28-s − 1.46·30-s − 0.179·31-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 20823 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20823 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(20823\) = \(3 \cdot 11 \cdot 631\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.32769\) |
| Root analytic conductor: | \(1.07343\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 20823,\ (\ :1/2, 1/2),\ 1)\) |
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$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) | |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) | ||
| 631 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) | ||
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) | 2.2.b_c |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.5.e_k | |
| 7 | $D_{4}$ | \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.7.d_n | |
| 13 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.13.d_p | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.17.d_bi | |
| 19 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.19.d_t | |
| 23 | $D_{4}$ | \( 1 + 5 T + 19 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.23.f_t | |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.29.a_bq | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.31.b_g | |
| 37 | $C_2^2$ | \( 1 + 3 T + 40 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.37.d_bo | |
| 41 | $D_{4}$ | \( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.41.ag_cy | |
| 43 | $D_{4}$ | \( 1 + 12 T + 108 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.43.m_ee | |
| 47 | $D_{4}$ | \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.47.ae_ak | |
| 53 | $D_{4}$ | \( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.53.ad_cs | |
| 59 | $D_{4}$ | \( 1 - T + 74 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.59.ab_cw | |
| 61 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.61.e_bu | |
| 67 | $D_{4}$ | \( 1 - 5 T + 87 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.67.af_dj | |
| 71 | $D_{4}$ | \( 1 - 3 T - T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.71.ad_ab | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.73.ag_ao | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.79.g_di | |
| 83 | $D_{4}$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.83.j_bo | |
| 89 | $D_{4}$ | \( 1 + 13 T + 187 T^{2} + 13 p T^{3} + p^{2} T^{4} \) | 2.89.n_hf | |
| 97 | $D_{4}$ | \( 1 + 5 T - 20 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.97.f_au | |
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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
−16.1893310011, −15.8653872802, −15.4033787421, −15.0461344810, −14.3352407718, −13.7622550173, −13.1301010771, −12.6204571376, −12.2330413205, −11.8068134095, −11.3134231127, −10.9828835885, −10.1962394008, −9.88764573430, −9.21157897798, −8.62029248512, −8.15975868665, −7.67032751558, −6.81237050364, −6.54461039182, −5.70322830644, −4.94370022091, −4.28908677885, −3.70291637114, −2.65860423411, 0, 0, 2.65860423411, 3.70291637114, 4.28908677885, 4.94370022091, 5.70322830644, 6.54461039182, 6.81237050364, 7.67032751558, 8.15975868665, 8.62029248512, 9.21157897798, 9.88764573430, 10.1962394008, 10.9828835885, 11.3134231127, 11.8068134095, 12.2330413205, 12.6204571376, 13.1301010771, 13.7622550173, 14.3352407718, 15.0461344810, 15.4033787421, 15.8653872802, 16.1893310011