# Properties

 Label 2-637-13.9-c1-0-12 Degree $2$ Conductor $637$ Sign $-0.0128 - 0.999i$ Analytic cond. $5.08647$ Root an. cond. $2.25532$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.30 + 2.26i)2-s + (−1.30 + 2.26i)3-s + (−2.42 − 4.20i)4-s − 2.61·5-s + (−3.42 − 5.93i)6-s + 7.47·8-s + (−1.92 − 3.33i)9-s + (3.42 − 5.93i)10-s + (−0.927 + 1.60i)11-s + 12.7·12-s + (2.5 − 2.59i)13-s + (3.42 − 5.93i)15-s + (−4.92 + 8.53i)16-s + (−0.736 − 1.27i)17-s + 10.0·18-s + (−0.927 − 1.60i)19-s + ⋯
 L(s)  = 1 + (−0.925 + 1.60i)2-s + (−0.755 + 1.30i)3-s + (−1.21 − 2.10i)4-s − 1.17·5-s + (−1.39 − 2.42i)6-s + 2.64·8-s + (−0.642 − 1.11i)9-s + (1.08 − 1.87i)10-s + (−0.279 + 0.484i)11-s + 3.66·12-s + (0.693 − 0.720i)13-s + (0.884 − 1.53i)15-s + (−1.23 + 2.13i)16-s + (−0.178 − 0.309i)17-s + 2.37·18-s + (−0.212 − 0.368i)19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$637$$    =    $$7^{2} \cdot 13$$ Sign: $-0.0128 - 0.999i$ Analytic conductor: $$5.08647$$ Root analytic conductor: $$2.25532$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{637} (295, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 637,\ (\ :1/2),\ -0.0128 - 0.999i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.222055 + 0.224921i$$ $$L(\frac12)$$ $$\approx$$ $$0.222055 + 0.224921i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad7 $$1$$
13 $$1 + (-2.5 + 2.59i)T$$
good2 $$1 + (1.30 - 2.26i)T + (-1 - 1.73i)T^{2}$$
3 $$1 + (1.30 - 2.26i)T + (-1.5 - 2.59i)T^{2}$$
5 $$1 + 2.61T + 5T^{2}$$
11 $$1 + (0.927 - 1.60i)T + (-5.5 - 9.52i)T^{2}$$
17 $$1 + (0.736 + 1.27i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (0.927 + 1.60i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (-2.23 + 3.87i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (3.54 - 6.14i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 - 4.70T + 31T^{2}$$
37 $$1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 + (-0.381 + 0.661i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (6.28 + 10.8i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 - 2.23T + 47T^{2}$$
53 $$1 - 3.76T + 53T^{2}$$
59 $$1 + (1.11 + 1.93i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-6.35 + 11.0i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + (-7.09 - 12.2i)T + (-35.5 + 61.4i)T^{2}$$
73 $$1 - 2T + 73T^{2}$$
79 $$1 - 4T + 79T^{2}$$
83 $$1 + 6.70T + 83T^{2}$$
89 $$1 + (-2.45 + 4.25i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + (-9.42 - 16.3i)T + (-48.5 + 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$