Dirichlet series
| L(s) = 1 | − 2-s − 3·3-s − 4-s − 3·5-s + 3·6-s − 2·7-s + 8-s + 2·9-s + 3·10-s − 5·11-s + 3·12-s − 3·13-s + 2·14-s + 9·15-s + 3·16-s − 3·17-s − 2·18-s − 4·19-s + 3·20-s + 6·21-s + 5·22-s + 3·23-s − 3·24-s + 2·25-s + 3·26-s + 6·27-s + 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 1/2·4-s − 1.34·5-s + 1.22·6-s − 0.755·7-s + 0.353·8-s + 2/3·9-s + 0.948·10-s − 1.50·11-s + 0.866·12-s − 0.832·13-s + 0.534·14-s + 2.32·15-s + 3/4·16-s − 0.727·17-s − 0.471·18-s − 0.917·19-s + 0.670·20-s + 1.30·21-s + 1.06·22-s + 0.625·23-s − 0.612·24-s + 2/5·25-s + 0.588·26-s + 1.15·27-s + 0.377·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 16190 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16190 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(16190\) = \(2 \cdot 5 \cdot 1619\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.03228\) |
| Root analytic conductor: | \(1.00797\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 16190,\ (\ :1/2, 1/2),\ 1)\) |
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$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) | ||
| 1619 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 36 T + p T^{2} ) \) | ||
| good | 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) | 2.3.d_h |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.7.c_c | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.11.f_w | |
| 13 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.13.d_h | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.17.d_bi | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.19.e_bm | |
| 23 | $C_2^2$ | \( 1 - 3 T + 26 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.23.ad_ba | |
| 29 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.29.f_bc | |
| 31 | $D_{4}$ | \( 1 + 7 T + 38 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.31.h_bm | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.37.b_acg | |
| 41 | $D_{4}$ | \( 1 - 7 T + 57 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.41.ah_cf | |
| 43 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.43.b_k | |
| 47 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) | 2.47.a_ba | |
| 53 | $D_{4}$ | \( 1 - T + 74 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.53.ab_cw | |
| 59 | $D_{4}$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.59.d_aw | |
| 61 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.61.m_fu | |
| 67 | $D_{4}$ | \( 1 + 9 T + 43 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.67.j_br | |
| 71 | $D_{4}$ | \( 1 - 15 T + 117 T^{2} - 15 p T^{3} + p^{2} T^{4} \) | 2.71.ap_en | |
| 73 | $D_{4}$ | \( 1 + 10 T + 42 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.73.k_bq | |
| 79 | $D_{4}$ | \( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.79.ag_co | |
| 83 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) | 2.83.a_abu | |
| 89 | $D_{4}$ | \( 1 + 3 T + 121 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.89.d_er | |
| 97 | $D_{4}$ | \( 1 - 5 T - 49 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.97.af_abx | |
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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
−16.5715364426, −16.2176143020, −15.6987040431, −15.0931743042, −14.9532848633, −14.0933569814, −13.2855643166, −12.8639860292, −12.4772217716, −12.0874926388, −11.4318943121, −10.9907630073, −10.6499400132, −10.2202523648, −9.30362545841, −9.04325486329, −8.20524571546, −7.72052138083, −7.22977131722, −6.42751617614, −5.75075359918, −5.23729903193, −4.61330673055, −3.77356829689, −2.72241808583, 0, 0, 2.72241808583, 3.77356829689, 4.61330673055, 5.23729903193, 5.75075359918, 6.42751617614, 7.22977131722, 7.72052138083, 8.20524571546, 9.04325486329, 9.30362545841, 10.2202523648, 10.6499400132, 10.9907630073, 11.4318943121, 12.0874926388, 12.4772217716, 12.8639860292, 13.2855643166, 14.0933569814, 14.9532848633, 15.0931743042, 15.6987040431, 16.2176143020, 16.5715364426