Dirichlet series
| L(s) = 1 | + 2-s − 2·3-s + 4-s − 3·5-s − 2·6-s − 7-s + 8-s − 3·10-s − 2·11-s − 2·12-s − 7·13-s − 14-s + 6·15-s + 16-s − 17-s − 3·20-s + 2·21-s − 2·22-s + 3·23-s − 2·24-s + 2·25-s − 7·26-s + 5·27-s − 28-s + 2·29-s + 6·30-s + 3·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 1.34·5-s − 0.816·6-s − 0.377·7-s + 0.353·8-s − 0.948·10-s − 0.603·11-s − 0.577·12-s − 1.94·13-s − 0.267·14-s + 1.54·15-s + 1/4·16-s − 0.242·17-s − 0.670·20-s + 0.436·21-s − 0.426·22-s + 0.625·23-s − 0.408·24-s + 2/5·25-s − 1.37·26-s + 0.962·27-s − 0.188·28-s + 0.371·29-s + 1.09·30-s + 0.538·31-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 12864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(12864\) = \(2^{6} \cdot 3 \cdot 67\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.820219\) |
| Root analytic conductor: | \(0.951661\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 12864,\ (\ :1/2, 1/2),\ -1)\) |
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 | $C_1$ | \( 1 - T \) | |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) | ||
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 11 T + p T^{2} ) \) | ||
| good | 5 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.5.d_h |
| 7 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) | 2.7.b_h | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.11.c_o | |
| 13 | $D_{4}$ | \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.13.h_be | |
| 17 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.17.b_g | |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) | 2.19.a_k | |
| 23 | $D_{4}$ | \( 1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.23.ad_n | |
| 29 | $D_{4}$ | \( 1 - 2 T - 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.29.ac_as | |
| 31 | $D_{4}$ | \( 1 - 3 T + 37 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.31.ad_bl | |
| 37 | $D_{4}$ | \( 1 + 7 T + 36 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.37.h_bk | |
| 41 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.41.c_k | |
| 43 | $D_{4}$ | \( 1 + 3 T + 59 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.43.d_ch | |
| 47 | $D_{4}$ | \( 1 - 5 T + 81 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.47.af_dd | |
| 53 | $D_{4}$ | \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.53.ab_abc | |
| 59 | $D_{4}$ | \( 1 + 5 T + 42 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.59.f_bq | |
| 61 | $D_{4}$ | \( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.61.ah_u | |
| 71 | $D_{4}$ | \( 1 - 3 T - 23 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.71.ad_ax | |
| 73 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) | 2.73.a_ek | |
| 79 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) | 2.79.a_ao | |
| 83 | $D_{4}$ | \( 1 - 11 T + 69 T^{2} - 11 p T^{3} + p^{2} T^{4} \) | 2.83.al_cr | |
| 89 | $D_{4}$ | \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.89.m_de | |
| 97 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.97.i_bq | |
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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
−16.4301477406, −16.0268389201, −15.5076987234, −15.1727954002, −14.7226138514, −14.0933590899, −13.5689925822, −12.8021162122, −12.4701928150, −11.9514712357, −11.7608393852, −11.2225634992, −10.6018829456, −10.2097806038, −9.49751778576, −8.63513252166, −7.96654736466, −7.39286087010, −6.90016894199, −6.28407617370, −5.37341105872, −4.98400820388, −4.39434380288, −3.38416229870, −2.57011148342, 0, 2.57011148342, 3.38416229870, 4.39434380288, 4.98400820388, 5.37341105872, 6.28407617370, 6.90016894199, 7.39286087010, 7.96654736466, 8.63513252166, 9.49751778576, 10.2097806038, 10.6018829456, 11.2225634992, 11.7608393852, 11.9514712357, 12.4701928150, 12.8021162122, 13.5689925822, 14.0933590899, 14.7226138514, 15.1727954002, 15.5076987234, 16.0268389201, 16.4301477406