Dirichlet series
| L(s) = 1 | + 3·3-s − 2·5-s + 7-s + 3·9-s + 6·11-s − 7·13-s − 6·15-s − 5·17-s + 8·19-s + 3·21-s + 2·23-s + 2·25-s + 4·29-s + 18·33-s − 2·35-s − 37-s − 21·39-s + 10·41-s − 7·43-s − 6·45-s + 5·47-s − 49-s − 15·51-s + 8·53-s − 12·55-s + 24·57-s + 4·59-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.894·5-s + 0.377·7-s + 9-s + 1.80·11-s − 1.94·13-s − 1.54·15-s − 1.21·17-s + 1.83·19-s + 0.654·21-s + 0.417·23-s + 2/5·25-s + 0.742·29-s + 3.13·33-s − 0.338·35-s − 0.164·37-s − 3.36·39-s + 1.56·41-s − 1.06·43-s − 0.894·45-s + 0.729·47-s − 1/7·49-s − 2.10·51-s + 1.09·53-s − 1.61·55-s + 3.17·57-s + 0.520·59-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 266240 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266240 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(266240\) = \(2^{12} \cdot 5 \cdot 13\) |
| Sign: | $1$ |
| Analytic conductor: | \(16.9756\) |
| Root analytic conductor: | \(2.02981\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 266240,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.896430058\) |
| \(L(\frac12)\) | \(\approx\) | \(2.896430058\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | ||
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) | ||
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 6 T + p T^{2} ) \) | ||
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) | 2.3.ad_g |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.7.ab_c | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) | 2.11.ag_w | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.17.f_u | |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) | 2.19.ai_cc | |
| 23 | $D_{4}$ | \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.23.ac_ag | |
| 29 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.29.ae_o | |
| 31 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) | 2.31.a_abi | |
| 37 | $D_{4}$ | \( 1 + T + 40 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.37.b_bo | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.41.ak_cg | |
| 43 | $D_{4}$ | \( 1 + 7 T + 38 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.43.h_bm | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.47.af_cg | |
| 53 | $D_{4}$ | \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.53.ai_dm | |
| 59 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.59.ae_be | |
| 61 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.61.k_cg | |
| 67 | $D_{4}$ | \( 1 - 6 T + 6 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.67.ag_g | |
| 71 | $D_{4}$ | \( 1 + T + 38 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.71.b_bm | |
| 73 | $D_{4}$ | \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.73.ag_da | |
| 79 | $D_{4}$ | \( 1 - 18 T + 222 T^{2} - 18 p T^{3} + p^{2} T^{4} \) | 2.79.as_io | |
| 83 | $D_{4}$ | \( 1 - 10 T + 38 T^{2} - 10 p T^{3} + p^{2} T^{4} \) | 2.83.ak_bm | |
| 89 | $D_{4}$ | \( 1 + 2 T - 38 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.89.c_abm | |
| 97 | $D_{4}$ | \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.97.ai_bi | |
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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
−13.4667519159, −12.6874269222, −12.2530705171, −11.9205901891, −11.6897187242, −11.1830754794, −10.6839936241, −9.99828828296, −9.47505187750, −9.32983918325, −8.86064836196, −8.55274013459, −7.95643835833, −7.52358054789, −7.31484837164, −6.72269820147, −6.27186696055, −5.21480258723, −4.87545430802, −4.24837832302, −3.74282360740, −3.23481705291, −2.59924102143, −2.12361273314, −0.970120138638, 0.970120138638, 2.12361273314, 2.59924102143, 3.23481705291, 3.74282360740, 4.24837832302, 4.87545430802, 5.21480258723, 6.27186696055, 6.72269820147, 7.31484837164, 7.52358054789, 7.95643835833, 8.55274013459, 8.86064836196, 9.32983918325, 9.47505187750, 9.99828828296, 10.6839936241, 11.1830754794, 11.6897187242, 11.9205901891, 12.2530705171, 12.6874269222, 13.4667519159