Properties

 Label 2-76-19.7-c5-0-5 Degree $2$ Conductor $76$ Sign $0.292 + 0.956i$ Analytic cond. $12.1891$ Root an. cond. $3.49129$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (−3.30 − 5.72i)3-s + (12.6 + 21.8i)5-s − 40.7·7-s + (99.6 − 172. i)9-s + 324.·11-s + (−48.8 + 84.6i)13-s + (83.3 − 144. i)15-s + (−1.05e3 − 1.83e3i)17-s + (806. − 1.35e3i)19-s + (134. + 232. i)21-s + (1.50e3 − 2.60e3i)23-s + (1.24e3 − 2.15e3i)25-s − 2.92e3·27-s + (−527. + 914. i)29-s + 7.60e3·31-s + ⋯
 L(s)  = 1 + (−0.211 − 0.367i)3-s + (0.225 + 0.390i)5-s − 0.314·7-s + (0.410 − 0.710i)9-s + 0.809·11-s + (−0.0801 + 0.138i)13-s + (0.0955 − 0.165i)15-s + (−0.887 − 1.53i)17-s + (0.512 − 0.858i)19-s + (0.0665 + 0.115i)21-s + (0.593 − 1.02i)23-s + (0.398 − 0.689i)25-s − 0.771·27-s + (−0.116 + 0.201i)29-s + 1.42·31-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$76$$    =    $$2^{2} \cdot 19$$ Sign: $0.292 + 0.956i$ Analytic conductor: $$12.1891$$ Root analytic conductor: $$3.49129$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{76} (45, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 76,\ (\ :5/2),\ 0.292 + 0.956i)$$

Particular Values

 $$L(3)$$ $$\approx$$ $$1.26519 - 0.936500i$$ $$L(\frac12)$$ $$\approx$$ $$1.26519 - 0.936500i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
19 $$1 + (-806. + 1.35e3i)T$$
good3 $$1 + (3.30 + 5.72i)T + (-121.5 + 210. i)T^{2}$$
5 $$1 + (-12.6 - 21.8i)T + (-1.56e3 + 2.70e3i)T^{2}$$
7 $$1 + 40.7T + 1.68e4T^{2}$$
11 $$1 - 324.T + 1.61e5T^{2}$$
13 $$1 + (48.8 - 84.6i)T + (-1.85e5 - 3.21e5i)T^{2}$$
17 $$1 + (1.05e3 + 1.83e3i)T + (-7.09e5 + 1.22e6i)T^{2}$$
23 $$1 + (-1.50e3 + 2.60e3i)T + (-3.21e6 - 5.57e6i)T^{2}$$
29 $$1 + (527. - 914. i)T + (-1.02e7 - 1.77e7i)T^{2}$$
31 $$1 - 7.60e3T + 2.86e7T^{2}$$
37 $$1 + 2.05e3T + 6.93e7T^{2}$$
41 $$1 + (-230. - 399. i)T + (-5.79e7 + 1.00e8i)T^{2}$$
43 $$1 + (188. + 326. i)T + (-7.35e7 + 1.27e8i)T^{2}$$
47 $$1 + (8.35e3 - 1.44e4i)T + (-1.14e8 - 1.98e8i)T^{2}$$
53 $$1 + (-5.23e3 + 9.06e3i)T + (-2.09e8 - 3.62e8i)T^{2}$$
59 $$1 + (-2.37e4 - 4.11e4i)T + (-3.57e8 + 6.19e8i)T^{2}$$
61 $$1 + (1.37e4 - 2.37e4i)T + (-4.22e8 - 7.31e8i)T^{2}$$
67 $$1 + (-1.72e4 + 2.98e4i)T + (-6.75e8 - 1.16e9i)T^{2}$$
71 $$1 + (-1.87e3 - 3.25e3i)T + (-9.02e8 + 1.56e9i)T^{2}$$
73 $$1 + (-6.14e3 - 1.06e4i)T + (-1.03e9 + 1.79e9i)T^{2}$$
79 $$1 + (-2.67e4 - 4.62e4i)T + (-1.53e9 + 2.66e9i)T^{2}$$
83 $$1 + 1.31e3T + 3.93e9T^{2}$$
89 $$1 + (-5.17e4 + 8.96e4i)T + (-2.79e9 - 4.83e9i)T^{2}$$
97 $$1 + (5.29e4 + 9.17e4i)T + (-4.29e9 + 7.43e9i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$