Dirichlet series
| L(s) = 1 | − 2-s + 4-s − 2·5-s + 2·7-s − 3·8-s − 2·9-s + 2·10-s − 3·11-s − 2·14-s + 16-s − 8·17-s + 2·18-s − 2·20-s + 3·22-s + 5·23-s − 2·25-s + 3·27-s + 2·28-s − 29-s − 5·31-s + 32-s + 8·34-s − 4·35-s − 2·36-s − 3·37-s + 6·40-s − 7·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.755·7-s − 1.06·8-s − 2/3·9-s + 0.632·10-s − 0.904·11-s − 0.534·14-s + 1/4·16-s − 1.94·17-s + 0.471·18-s − 0.447·20-s + 0.639·22-s + 1.04·23-s − 2/5·25-s + 0.577·27-s + 0.377·28-s − 0.185·29-s − 0.898·31-s + 0.176·32-s + 1.37·34-s − 0.676·35-s − 1/3·36-s − 0.493·37-s + 0.948·40-s − 1.09·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 19119 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19119 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(19119\) = \(3 \cdot 6373\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.21904\) |
| Root analytic conductor: | \(1.05076\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 19119,\ (\ :1/2, 1/2),\ -1)\) |
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$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) | |
| 6373 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 70 T + p T^{2} ) \) | ||
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) | 2.2.b_a |
| 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.5.c_g | |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.7.ac_c | |
| 11 | $D_{4}$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.11.d_c | |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) | 2.13.a_g | |
| 17 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.17.i_bm | |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) | 2.19.a_k | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) | 2.23.af_bu | |
| 29 | $D_{4}$ | \( 1 + T + 48 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.29.b_bw | |
| 31 | $D_{4}$ | \( 1 + 5 T + 46 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.31.f_bu | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.37.d_e | |
| 41 | $D_{4}$ | \( 1 + 7 T + 72 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.41.h_cu | |
| 43 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) | 2.43.a_as | |
| 47 | $D_{4}$ | \( 1 + T - 18 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.47.b_as | |
| 53 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.53.e_bm | |
| 59 | $D_{4}$ | \( 1 - 7 T + 34 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.59.ah_bi | |
| 61 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.61.ai_da | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) | 2.67.m_cs | |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) | 2.71.a_aby | |
| 73 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.73.e_be | |
| 79 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) | 2.79.am_es | |
| 83 | $D_{4}$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.83.af_ag | |
| 89 | $D_{4}$ | \( 1 + 6 T + 118 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.89.g_eo | |
| 97 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.97.ae_bu | |
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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
−15.8829803069, −15.6049492429, −15.1704936161, −14.8951058044, −14.2665221931, −13.6007044317, −13.1514436915, −12.6265146736, −11.9083016239, −11.5057264659, −11.2495517637, −10.7355528570, −10.2498104083, −9.32738808688, −8.86901192056, −8.55224298716, −8.01921238311, −7.39937821320, −6.81896046581, −6.23641606959, −5.28740045303, −4.81468897201, −3.86565861774, −2.92427391676, −2.12190135350, 0, 2.12190135350, 2.92427391676, 3.86565861774, 4.81468897201, 5.28740045303, 6.23641606959, 6.81896046581, 7.39937821320, 8.01921238311, 8.55224298716, 8.86901192056, 9.32738808688, 10.2498104083, 10.7355528570, 11.2495517637, 11.5057264659, 11.9083016239, 12.6265146736, 13.1514436915, 13.6007044317, 14.2665221931, 14.8951058044, 15.1704936161, 15.6049492429, 15.8829803069