Dirichlet series
| L(s) = 1 | − 2-s − 2·3-s − 4-s − 3·5-s + 2·6-s − 4·7-s + 3·8-s − 2·9-s + 3·10-s + 2·11-s + 2·12-s + 5·13-s + 4·14-s + 6·15-s − 16-s + 4·17-s + 2·18-s − 6·19-s + 3·20-s + 8·21-s − 2·22-s − 6·23-s − 6·24-s + 2·25-s − 5·26-s + 10·27-s + 4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s − 1.34·5-s + 0.816·6-s − 1.51·7-s + 1.06·8-s − 2/3·9-s + 0.948·10-s + 0.603·11-s + 0.577·12-s + 1.38·13-s + 1.06·14-s + 1.54·15-s − 1/4·16-s + 0.970·17-s + 0.471·18-s − 1.37·19-s + 0.670·20-s + 1.74·21-s − 0.426·22-s − 1.25·23-s − 1.22·24-s + 2/5·25-s − 0.980·26-s + 1.92·27-s + 0.755·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 2080 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2080 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(2080\) = \(2^{5} \cdot 5 \cdot 13\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.132622\) |
| Root analytic conductor: | \(0.603468\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 2080,\ (\ :1/2, 1/2),\ -1)\) |
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 | $C_2$ | \( 1 + T + p T^{2} \) | |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) | ||
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) | ||
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.3.c_g |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.7.e_o | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) | 2.11.ac_w | |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.17.ae_bm | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.19.g_bm | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.23.g_bu | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.29.i_bm | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.31.k_ck | |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) | 2.37.e_da | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.41.ae_w | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) | 2.43.ak_di | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) | 2.47.ae_dq | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.53.aq_fe | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) | 2.59.ag_eo | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.61.i_dy | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.67.e_fe | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) | 2.71.ag_fm | |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) | 2.73.m_ha | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.79.m_gc | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) | 2.83.q_gk | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.89.am_hq | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.97.aq_gc | |
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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
−19.1887245783, −18.4590779879, −18.1183629461, −17.3405617440, −16.7385636699, −16.6814250399, −15.9784997767, −15.7488207479, −14.5765256398, −14.3654180706, −13.2768755254, −12.9146361946, −12.1373187793, −11.7666126827, −10.9076921437, −10.7527712458, −9.84497658306, −8.95538623117, −8.67915048394, −7.77199473906, −7.04231773997, −5.87146418849, −5.80677556176, −4.03550878859, −3.67478222653, 0, 3.67478222653, 4.03550878859, 5.80677556176, 5.87146418849, 7.04231773997, 7.77199473906, 8.67915048394, 8.95538623117, 9.84497658306, 10.7527712458, 10.9076921437, 11.7666126827, 12.1373187793, 12.9146361946, 13.2768755254, 14.3654180706, 14.5765256398, 15.7488207479, 15.9784997767, 16.6814250399, 16.7385636699, 17.3405617440, 18.1183629461, 18.4590779879, 19.1887245783