Properties

 Label 2-756-36.23-c1-0-13 Degree $2$ Conductor $756$ Sign $0.730 + 0.682i$ Analytic cond. $6.03669$ Root an. cond. $2.45696$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (0.0405 − 1.41i)2-s + (−1.99 − 0.114i)4-s + (2.16 + 1.25i)5-s + (0.866 − 0.5i)7-s + (−0.243 + 2.81i)8-s + (1.85 − 3.01i)10-s + (0.351 + 0.608i)11-s + (1.55 − 2.69i)13-s + (−0.671 − 1.24i)14-s + (3.97 + 0.458i)16-s + 7.91i·17-s + 2.37i·19-s + (−4.18 − 2.74i)20-s + (0.874 − 0.472i)22-s + (0.346 − 0.600i)23-s + ⋯
 L(s)  = 1 + (0.0287 − 0.999i)2-s + (−0.998 − 0.0573i)4-s + (0.969 + 0.559i)5-s + (0.327 − 0.188i)7-s + (−0.0860 + 0.996i)8-s + (0.587 − 0.953i)10-s + (0.105 + 0.183i)11-s + (0.430 − 0.746i)13-s + (−0.179 − 0.332i)14-s + (0.993 + 0.114i)16-s + 1.91i·17-s + 0.544i·19-s + (−0.935 − 0.614i)20-s + (0.186 − 0.100i)22-s + (0.0722 − 0.125i)23-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 + 0.682i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.730 + 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$756$$    =    $$2^{2} \cdot 3^{3} \cdot 7$$ Sign: $0.730 + 0.682i$ Analytic conductor: $$6.03669$$ Root analytic conductor: $$2.45696$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{756} (71, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 756,\ (\ :1/2),\ 0.730 + 0.682i)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$1.63096 - 0.643643i$$ $$L(\frac12)$$ $$\approx$$ $$1.63096 - 0.643643i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.0405 + 1.41i)T$$
3 $$1$$
7 $$1 + (-0.866 + 0.5i)T$$
good5 $$1 + (-2.16 - 1.25i)T + (2.5 + 4.33i)T^{2}$$
11 $$1 + (-0.351 - 0.608i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 + (-1.55 + 2.69i)T + (-6.5 - 11.2i)T^{2}$$
17 $$1 - 7.91iT - 17T^{2}$$
19 $$1 - 2.37iT - 19T^{2}$$
23 $$1 + (-0.346 + 0.600i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (-8.70 + 5.02i)T + (14.5 - 25.1i)T^{2}$$
31 $$1 + (-7.34 - 4.24i)T + (15.5 + 26.8i)T^{2}$$
37 $$1 - 2.85T + 37T^{2}$$
41 $$1 + (5.82 + 3.36i)T + (20.5 + 35.5i)T^{2}$$
43 $$1 + (-3.17 + 1.83i)T + (21.5 - 37.2i)T^{2}$$
47 $$1 + (3.67 + 6.37i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 - 0.889iT - 53T^{2}$$
59 $$1 + (3.63 - 6.28i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (2.39 + 4.15i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (7.40 + 4.27i)T + (33.5 + 58.0i)T^{2}$$
71 $$1 + 6.34T + 71T^{2}$$
73 $$1 - 8.20T + 73T^{2}$$
79 $$1 + (-2.27 + 1.31i)T + (39.5 - 68.4i)T^{2}$$
83 $$1 + (-6.50 - 11.2i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 - 1.71iT - 89T^{2}$$
97 $$1 + (6.81 + 11.7i)T + (-48.5 + 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$