Dirichlet series
| L(s) = 1 | − 3-s − 4-s − 2·7-s + 2·11-s + 12-s − 5·13-s − 3·16-s + 2·17-s + 4·19-s + 2·21-s + 6·23-s − 5·25-s + 4·27-s + 2·28-s + 4·29-s + 4·31-s − 2·33-s + 2·37-s + 5·39-s − 6·41-s − 6·43-s − 2·44-s − 8·47-s + 3·48-s − 2·49-s − 2·51-s + 5·52-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s − 0.755·7-s + 0.603·11-s + 0.288·12-s − 1.38·13-s − 3/4·16-s + 0.485·17-s + 0.917·19-s + 0.436·21-s + 1.25·23-s − 25-s + 0.769·27-s + 0.377·28-s + 0.742·29-s + 0.718·31-s − 0.348·33-s + 0.328·37-s + 0.800·39-s − 0.937·41-s − 0.914·43-s − 0.301·44-s − 1.16·47-s + 0.433·48-s − 2/7·49-s − 0.280·51-s + 0.693·52-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(975\) = \(3 \cdot 5^{2} \cdot 13\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0621668\) |
| Root analytic conductor: | \(0.499332\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 975,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.3987858446\) |
| \(L(\frac12)\) | \(\approx\) | \(0.3987858446\) |
| \(L(\frac{3}{2})\) | 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 | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) | |
| 5 | $C_2$ | \( 1 + p T^{2} \) | ||
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) | ||
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) | 2.2.a_b |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.7.c_g | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.11.ac_ac | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.17.ac_ao | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.19.ae_g | |
| 23 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.23.ag_w | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.29.ae_ac | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.31.ae_be | |
| 37 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.37.ac_c | |
| 41 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.41.g_bi | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.43.g_bu | |
| 47 | $D_{4}$ | \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.47.i_de | |
| 53 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.53.ac_bi | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.59.c_dq | |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.61.a_w | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.67.c_cc | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.71.ac_dq | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.73.c_co | |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.79.a_dq | |
| 83 | $D_{4}$ | \( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} \) | 2.83.am_de | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.89.g_ec | |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) | 2.97.k_by | |
| 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
−19.3063585959, −19.1415602959, −18.2259323823, −17.7908318312, −17.2639085110, −16.6630725064, −16.2824956198, −15.4915962947, −14.9049769793, −14.1875069081, −13.6228164916, −13.0184463465, −12.2003267383, −11.8465758768, −11.1501600679, −10.0291431575, −9.76490738467, −9.03038832598, −8.08310694870, −7.03174084463, −6.49691372712, −5.30564381608, −4.59858608848, −3.11895266887, 3.11895266887, 4.59858608848, 5.30564381608, 6.49691372712, 7.03174084463, 8.08310694870, 9.03038832598, 9.76490738467, 10.0291431575, 11.1501600679, 11.8465758768, 12.2003267383, 13.0184463465, 13.6228164916, 14.1875069081, 14.9049769793, 15.4915962947, 16.2824956198, 16.6630725064, 17.2639085110, 17.7908318312, 18.2259323823, 19.1415602959, 19.3063585959