Dirichlet series
| L(s) = 1 | − 4-s + 2·5-s − 5·7-s − 2·9-s + 6·11-s − 4·13-s − 3·16-s + 17-s − 3·19-s − 2·20-s + 6·23-s + 2·25-s + 5·28-s + 3·31-s − 10·35-s + 2·36-s − 5·37-s + 4·41-s + 5·43-s − 6·44-s − 4·45-s + 7·49-s + 4·52-s − 6·53-s + 12·55-s + 11·59-s + 4·61-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.894·5-s − 1.88·7-s − 2/3·9-s + 1.80·11-s − 1.10·13-s − 3/4·16-s + 0.242·17-s − 0.688·19-s − 0.447·20-s + 1.25·23-s + 2/5·25-s + 0.944·28-s + 0.538·31-s − 1.69·35-s + 1/3·36-s − 0.821·37-s + 0.624·41-s + 0.762·43-s − 0.904·44-s − 0.596·45-s + 49-s + 0.554·52-s − 0.824·53-s + 1.61·55-s + 1.43·59-s + 0.512·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 1795 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1795 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1795\) = \(5 \cdot 359\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.114450\) |
| Root analytic conductor: | \(0.581640\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 1795,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.5666129195\) |
| \(L(\frac12)\) | \(\approx\) | \(0.5666129195\) |
| \(L(\frac{3}{2})\) | 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 | 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) | |
| 359 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 12 T + p T^{2} ) \) | ||
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) | 2.2.a_b |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.3.a_c | |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.7.f_s | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) | 2.11.ag_w | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.13.e_ba | |
| 17 | $D_{4}$ | \( 1 - T - 20 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.17.ab_au | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.19.d_k | |
| 23 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.23.ag_bi | |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.29.a_w | |
| 31 | $D_{4}$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.31.ad_ac | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.37.f_ci | |
| 41 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.41.ae_k | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.43.af_ck | |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) | 2.47.a_ao | |
| 53 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.53.g_w | |
| 59 | $D_{4}$ | \( 1 - 11 T + 82 T^{2} - 11 p T^{3} + p^{2} T^{4} \) | 2.59.al_de | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.61.ae_as | |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) | 2.67.a_afe | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.71.ab_cs | |
| 73 | $D_{4}$ | \( 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.73.h_u | |
| 79 | $D_{4}$ | \( 1 - 14 T + 170 T^{2} - 14 p T^{3} + p^{2} T^{4} \) | 2.79.ao_go | |
| 83 | $D_{4}$ | \( 1 + 5 T - 50 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.83.f_aby | |
| 89 | $D_{4}$ | \( 1 + 4 T + 142 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.89.e_fm | |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.97.a_dq | |
| show more | ||||
| show less | ||||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.2240088262, −18.6804174010, −17.6657623089, −17.3381290532, −17.0563406083, −16.4246506445, −15.9335438192, −15.0305416403, −14.4855203486, −14.0690881747, −13.4369598866, −12.8486026148, −12.4387802037, −11.6991315572, −10.9479864712, −9.93809545339, −9.64197443449, −9.08939964627, −8.65783135488, −7.14622333375, −6.58038419211, −6.08184174277, −4.99322781938, −3.83902026399, −2.70513678386, 2.70513678386, 3.83902026399, 4.99322781938, 6.08184174277, 6.58038419211, 7.14622333375, 8.65783135488, 9.08939964627, 9.64197443449, 9.93809545339, 10.9479864712, 11.6991315572, 12.4387802037, 12.8486026148, 13.4369598866, 14.0690881747, 14.4855203486, 15.0305416403, 15.9335438192, 16.4246506445, 17.0563406083, 17.3381290532, 17.6657623089, 18.6804174010, 19.2240088262