Dirichlet series
| L(χ,s) = 1 | + (−0.623 + 0.781i)2-s + (0.330 + 0.943i)3-s + (−0.222 − 0.974i)4-s + (−0.330 + 0.943i)5-s + (−0.943 − 0.330i)6-s + (−0.900 − 0.433i)7-s + (0.900 + 0.433i)8-s + (−0.781 + 0.623i)9-s + (−0.532 − 0.846i)10-s + (−0.974 − 0.222i)11-s + (0.846 − 0.532i)12-s + (−0.433 + 0.900i)13-s + (0.900 − 0.433i)14-s − 15-s + (−0.900 + 0.433i)16-s + (0.993 − 0.111i)17-s + ⋯ |
| L(s,χ) = 1 | + (−0.623 + 0.781i)2-s + (0.330 + 0.943i)3-s + (−0.222 − 0.974i)4-s + (−0.330 + 0.943i)5-s + (−0.943 − 0.330i)6-s + (−0.900 − 0.433i)7-s + (0.900 + 0.433i)8-s + (−0.781 + 0.623i)9-s + (−0.532 − 0.846i)10-s + (−0.974 − 0.222i)11-s + (0.846 − 0.532i)12-s + (−0.433 + 0.900i)13-s + (0.900 − 0.433i)14-s − 15-s + (−0.900 + 0.433i)16-s + (0.993 − 0.111i)17-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(\chi,s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr
=\mathstrut & (-0.940 - 0.340i)\, \Lambda(\overline{\chi},1-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s,\chi)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr
=\mathstrut & (-0.940 - 0.340i)\, \Lambda(1-s,\overline{\chi})
\end{aligned}
\]
Invariants
| \( d \) | = | \(1\) |
| \( N \) | = | \(113\) |
| \( \varepsilon \) | = | $-0.940 - 0.340i$ |
| motivic weight | = | \(0\) |
| character | : | $\chi_{113} (13, \cdot )$ |
| Sato-Tate | : | $\mu(56)$ |
| primitive | : | yes |
| self-dual | : | no |
| analytic rank | = | 0 |
| Selberg data | = | $(1,\ 113,\ (0:\ ),\ -0.940 - 0.340i)$ |
| $L(\chi,\frac{1}{2})$ | $\approx$ | $-0.08028866876 + 0.4576198386i$ |
| $L(\frac12,\chi)$ | $\approx$ | $-0.08028866876 + 0.4576198386i$ |
| $L(\chi,1)$ | $\approx$ | 0.3804526868 + 0.4693711288i |
| $L(1,\chi)$ | $\approx$ | 0.3804526868 + 0.4693711288i |
Euler product
\[\begin{aligned}
L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]
\[\begin{aligned}
L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]