Dirichlet series
| L(s) = 1 | − 3-s − 2·4-s + 5-s − 2·7-s − 9-s + 7·11-s + 2·12-s + 3·13-s − 15-s + 4·16-s − 2·19-s − 2·20-s + 2·21-s + 2·23-s − 3·25-s + 4·28-s + 3·29-s − 13·31-s − 7·33-s − 2·35-s + 2·36-s − 3·39-s − 2·41-s + 3·43-s − 14·44-s − 45-s − 8·47-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.447·5-s − 0.755·7-s − 1/3·9-s + 2.11·11-s + 0.577·12-s + 0.832·13-s − 0.258·15-s + 16-s − 0.458·19-s − 0.447·20-s + 0.436·21-s + 0.417·23-s − 3/5·25-s + 0.755·28-s + 0.557·29-s − 2.33·31-s − 1.21·33-s − 0.338·35-s + 1/3·36-s − 0.480·39-s − 0.312·41-s + 0.457·43-s − 2.11·44-s − 0.149·45-s − 1.16·47-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 348928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(348928\) = \(2^{8} \cdot 29 \cdot 47\) |
| Sign: | $-1$ |
| Analytic conductor: | \(22.2479\) |
| Root analytic conductor: | \(2.17181\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 348928,\ (\ :1/2, 1/2),\ -1)\) |
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 | $C_2$ | \( 1 + p T^{2} \) | |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) | ||
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 9 T + p T^{2} ) \) | ||
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.3.b_c |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.5.ab_e | |
| 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$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.11.ah_bg | |
| 13 | $C_2^2$ | \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.13.ad_ae | |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) | 2.17.a_o | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.19.c_o | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.23.ac_ac | |
| 31 | $D_{4}$ | \( 1 + 13 T + 94 T^{2} + 13 p T^{3} + p^{2} T^{4} \) | 2.31.n_dq | |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) | 2.37.a_c | |
| 41 | $D_{4}$ | \( 1 + 2 T - 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.41.c_abe | |
| 43 | $D_{4}$ | \( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.43.ad_ca | |
| 53 | $D_{4}$ | \( 1 + 9 T + 98 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.53.j_du | |
| 59 | $D_{4}$ | \( 1 - 2 T - 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.59.ac_abi | |
| 61 | $D_{4}$ | \( 1 + 14 T + 138 T^{2} + 14 p T^{3} + p^{2} T^{4} \) | 2.61.o_fi | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.67.ai_g | |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.71.a_ec | |
| 73 | $D_{4}$ | \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.73.ai_dm | |
| 79 | $D_{4}$ | \( 1 + 7 T + 110 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.79.h_eg | |
| 83 | $D_{4}$ | \( 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.83.ag_bu | |
| 89 | $D_{4}$ | \( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.89.ae_fe | |
| 97 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.97.m_dy | |
| 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
−12.9606959382, −12.8664193145, −12.4008250293, −11.9203952227, −11.5583333356, −10.9987568332, −10.8306091513, −10.1814900532, −9.58544603070, −9.41901946029, −9.04369971322, −8.74261205410, −8.12148677693, −7.64735512775, −6.80760243008, −6.57733453074, −6.15055916443, −5.64921642102, −5.31088227815, −4.50679907427, −4.02838632705, −3.57539077253, −3.11985511314, −1.85153051047, −1.22425396196, 0, 1.22425396196, 1.85153051047, 3.11985511314, 3.57539077253, 4.02838632705, 4.50679907427, 5.31088227815, 5.64921642102, 6.15055916443, 6.57733453074, 6.80760243008, 7.64735512775, 8.12148677693, 8.74261205410, 9.04369971322, 9.41901946029, 9.58544603070, 10.1814900532, 10.8306091513, 10.9987568332, 11.5583333356, 11.9203952227, 12.4008250293, 12.8664193145, 12.9606959382