Dirichlet series
| L(s) = 1 | − 3-s − 3·5-s − 2·9-s − 4·11-s + 8·13-s + 3·15-s − 8·19-s − 4·23-s + 2·25-s + 2·27-s − 16·29-s + 4·33-s − 8·39-s + 4·41-s − 4·43-s + 6·45-s + 12·47-s − 14·49-s + 8·53-s + 12·55-s + 8·57-s − 12·59-s + 4·61-s − 24·65-s + 12·67-s + 4·69-s − 4·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 2/3·9-s − 1.20·11-s + 2.21·13-s + 0.774·15-s − 1.83·19-s − 0.834·23-s + 2/5·25-s + 0.384·27-s − 2.97·29-s + 0.696·33-s − 1.28·39-s + 0.624·41-s − 0.609·43-s + 0.894·45-s + 1.75·47-s − 2·49-s + 1.09·53-s + 1.61·55-s + 1.05·57-s − 1.56·59-s + 0.512·61-s − 2.97·65-s + 1.46·67-s + 0.481·69-s − 0.468·73-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 15360 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15360 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(15360\) = \(2^{10} \cdot 3 \cdot 5\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.979366\) |
| Root analytic conductor: | \(0.994801\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 15360,\ (\ :1/2, 1/2),\ -1)\) |
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 | \( 1 \) | ||
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) | ||
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) | ||
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.7.a_o |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.11.e_w | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.13.ai_bm | |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.17.a_be | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.19.i_bm | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.23.e_bu | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.29.q_eo | |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.31.a_ck | |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.37.a_cs | |
| 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$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.43.e_di | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) | 2.47.am_dq | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.53.ai_w | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.59.m_eo | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.61.ae_as | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) | 2.67.am_fe | |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.71.a_fm | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.73.e_fe | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) | 2.79.ai_gc | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) | 2.83.ae_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$ | \( ( 1 - 18 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.97.ae_acg | |
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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
−16.2652641967, −15.7488207479, −15.5114390469, −14.9757145558, −14.5765256398, −13.7392025398, −13.2768755254, −12.9418318979, −12.3574631320, −11.7666126827, −11.1766293089, −10.9076921437, −10.7974752774, −9.83549734579, −8.95538623117, −8.58268357776, −7.88889261724, −7.77199473906, −6.76366111245, −5.87146418849, −5.86487938441, −4.78210199118, −3.76484405046, −3.67478222653, −2.15654793578, 0, 2.15654793578, 3.67478222653, 3.76484405046, 4.78210199118, 5.86487938441, 5.87146418849, 6.76366111245, 7.77199473906, 7.88889261724, 8.58268357776, 8.95538623117, 9.83549734579, 10.7974752774, 10.9076921437, 11.1766293089, 11.7666126827, 12.3574631320, 12.9418318979, 13.2768755254, 13.7392025398, 14.5765256398, 14.9757145558, 15.5114390469, 15.7488207479, 16.2652641967