Dirichlet series
| L(s) = 1 | − 3·3-s + 4-s + 3·9-s − 6·11-s − 3·12-s + 16-s − 3·17-s − 7·19-s + 2·23-s + 25-s − 6·29-s + 4·31-s + 18·33-s + 3·36-s + 4·37-s + 3·41-s + 2·43-s − 6·44-s − 8·47-s − 3·48-s − 2·49-s + 9·51-s − 12·53-s + 21·57-s − 9·59-s + 64-s − 18·67-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1/2·4-s + 9-s − 1.80·11-s − 0.866·12-s + 1/4·16-s − 0.727·17-s − 1.60·19-s + 0.417·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s + 3.13·33-s + 1/2·36-s + 0.657·37-s + 0.468·41-s + 0.304·43-s − 0.904·44-s − 1.16·47-s − 0.433·48-s − 2/7·49-s + 1.26·51-s − 1.64·53-s + 2.78·57-s − 1.17·59-s + 1/8·64-s − 2.19·67-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 14564 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14564 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(14564\) = \(2^{2} \cdot 11 \cdot 331\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.928613\) |
| Root analytic conductor: | \(0.981654\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 14564,\ (\ :1/2, 1/2),\ -1)\) |
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_1$ | \( ( 1 - T )( 1 + T ) \) | |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 5 T + p T^{2} ) \) | ||
| 331 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 10 T + p T^{2} ) \) | ||
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) | 2.3.d_g |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) | 2.5.a_ab | |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) | 2.7.a_c | |
| 13 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) | 2.13.a_h | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.17.d_bi | |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.19.h_be | |
| 23 | $D_{4}$ | \( 1 - 2 T - 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.23.ac_ak | |
| 29 | $D_{4}$ | \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.29.g_br | |
| 31 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.31.ae_bm | |
| 37 | $D_{4}$ | \( 1 - 4 T + 72 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.37.ae_cu | |
| 41 | $D_{4}$ | \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.41.ad_bg | |
| 43 | $D_{4}$ | \( 1 - 2 T - 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.43.ac_as | |
| 47 | $D_{4}$ | \( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.47.i_cg | |
| 53 | $D_{4}$ | \( 1 + 12 T + 113 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.53.m_ej | |
| 59 | $D_{4}$ | \( 1 + 9 T + 70 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.59.j_cs | |
| 61 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) | 2.61.a_de | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) | 2.67.s_hr | |
| 71 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.71.ae_k | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.73.c_es | |
| 79 | $D_{4}$ | \( 1 - 12 T + 168 T^{2} - 12 p T^{3} + p^{2} T^{4} \) | 2.79.am_gm | |
| 83 | $D_{4}$ | \( 1 + 3 T + 92 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.83.d_do | |
| 89 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.89.ae_dm | |
| 97 | $D_{4}$ | \( 1 - 15 T + 120 T^{2} - 15 p T^{3} + p^{2} T^{4} \) | 2.97.ap_eq | |
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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
−16.4193893349, −16.0177034428, −15.5154279865, −15.0600081417, −14.6994743208, −13.7899565364, −13.2114154563, −12.7662483626, −12.5541408278, −11.7433473404, −11.3267394492, −10.8260564937, −10.7375856544, −10.1257121750, −9.34698418334, −8.62372948043, −7.87003997020, −7.49330049673, −6.50883014130, −6.22088968936, −5.67656862705, −4.90547014058, −4.52733560238, −3.11607122437, −2.12789563794, 0, 2.12789563794, 3.11607122437, 4.52733560238, 4.90547014058, 5.67656862705, 6.22088968936, 6.50883014130, 7.49330049673, 7.87003997020, 8.62372948043, 9.34698418334, 10.1257121750, 10.7375856544, 10.8260564937, 11.3267394492, 11.7433473404, 12.5541408278, 12.7662483626, 13.2114154563, 13.7899565364, 14.6994743208, 15.0600081417, 15.5154279865, 16.0177034428, 16.4193893349