Dirichlet series
| L(χ,s) = 1 | + (−0.891 + 0.452i)2-s + (0.935 + 0.353i)3-s + (0.590 − 0.806i)4-s + (−0.968 − 0.250i)5-s + (−0.994 + 0.108i)6-s + (−0.999 + 0.0361i)7-s + (−0.161 + 0.986i)8-s + (0.750 + 0.661i)9-s + (0.976 − 0.214i)10-s + (0.907 − 0.419i)11-s + (0.837 − 0.546i)12-s + (−0.0180 + 0.999i)13-s + (0.874 − 0.484i)14-s + (−0.817 − 0.576i)15-s + (−0.302 − 0.953i)16-s + (−0.161 − 0.986i)17-s + ⋯ |
| L(s,χ) = 1 | + (−0.891 + 0.452i)2-s + (0.935 + 0.353i)3-s + (0.590 − 0.806i)4-s + (−0.968 − 0.250i)5-s + (−0.994 + 0.108i)6-s + (−0.999 + 0.0361i)7-s + (−0.161 + 0.986i)8-s + (0.750 + 0.661i)9-s + (0.976 − 0.214i)10-s + (0.907 − 0.419i)11-s + (0.837 − 0.546i)12-s + (−0.0180 + 0.999i)13-s + (0.874 − 0.484i)14-s + (−0.817 − 0.576i)15-s + (−0.302 − 0.953i)16-s + (−0.161 − 0.986i)17-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(\chi,s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr
=\mathstrut & (0.318 + 0.948i)\, \Lambda(\overline{\chi},1-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s,\chi)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr
=\mathstrut & (0.318 + 0.948i)\, \Lambda(1-s,\overline{\chi})
\end{aligned}
\]
Invariants
| \( d \) | = | \(1\) |
| \( N \) | = | \(349\) |
| \( \varepsilon \) | = | $0.318 + 0.948i$ |
| motivic weight | = | \(0\) |
| character | : | $\chi_{349} (116, \cdot )$ |
| Sato-Tate | : | $\mu(87)$ |
| primitive | : | yes |
| self-dual | : | no |
| analytic rank | = | 0 |
| Selberg data | = | $(1,\ 349,\ (0:\ ),\ 0.318 + 0.948i)$ |
| $L(\chi,\frac{1}{2})$ | $\approx$ | $0.7404921708 + 0.5325382673i$ |
| $L(\frac12,\chi)$ | $\approx$ | $0.7404921708 + 0.5325382673i$ |
| $L(\chi,1)$ | $\approx$ | 0.7744951065 + 0.2735363169i |
| $L(1,\chi)$ | $\approx$ | 0.7744951065 + 0.2735363169i |
Euler product
\[\begin{aligned}
L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]
\[\begin{aligned}
L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]