# Properties

 Label 2-637-91.4-c1-0-29 Degree $2$ Conductor $637$ Sign $-0.923 + 0.384i$ 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 − 1.27i·2-s + (−0.583 − 1.01i)3-s + 0.370·4-s + (−1.57 + 0.907i)5-s + (−1.29 + 0.745i)6-s − 3.02i·8-s + (0.817 − 1.41i)9-s + (1.15 + 2.00i)10-s + (2.40 − 1.38i)11-s + (−0.216 − 0.374i)12-s + (3.58 − 0.402i)13-s + (1.83 + 1.05i)15-s − 3.12·16-s + 2.74·17-s + (−1.80 − 1.04i)18-s + (−5.08 − 2.93i)19-s + ⋯
 L(s)  = 1 − 0.902i·2-s + (−0.337 − 0.583i)3-s + 0.185·4-s + (−0.702 + 0.405i)5-s + (−0.527 + 0.304i)6-s − 1.06i·8-s + (0.272 − 0.472i)9-s + (0.366 + 0.634i)10-s + (0.725 − 0.418i)11-s + (−0.0624 − 0.108i)12-s + (0.993 − 0.111i)13-s + (0.473 + 0.273i)15-s − 0.780·16-s + 0.665·17-s + (−0.426 − 0.246i)18-s + (−1.16 − 0.673i)19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$637$$    =    $$7^{2} \cdot 13$$ Sign: $-0.923 + 0.384i$ Analytic conductor: $$5.08647$$ Root analytic conductor: $$2.25532$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{637} (459, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 637,\ (\ :1/2),\ -0.923 + 0.384i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.252077 - 1.26071i$$ $$L(\frac12)$$ $$\approx$$ $$0.252077 - 1.26071i$$ $$L(\frac{3}{2})$$ 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 + (-3.58 + 0.402i)T$$
good2 $$1 + 1.27iT - 2T^{2}$$
3 $$1 + (0.583 + 1.01i)T + (-1.5 + 2.59i)T^{2}$$
5 $$1 + (1.57 - 0.907i)T + (2.5 - 4.33i)T^{2}$$
11 $$1 + (-2.40 + 1.38i)T + (5.5 - 9.52i)T^{2}$$
17 $$1 - 2.74T + 17T^{2}$$
19 $$1 + (5.08 + 2.93i)T + (9.5 + 16.4i)T^{2}$$
23 $$1 + 6.99T + 23T^{2}$$
29 $$1 + (-1.75 + 3.04i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + (1.79 + 1.03i)T + (15.5 + 26.8i)T^{2}$$
37 $$1 + 1.74iT - 37T^{2}$$
41 $$1 + (5.51 + 3.18i)T + (20.5 + 35.5i)T^{2}$$
43 $$1 + (-4.55 - 7.88i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + (-5.76 + 3.32i)T + (23.5 - 40.7i)T^{2}$$
53 $$1 + (-5.24 + 9.08i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + 3.07iT - 59T^{2}$$
61 $$1 + (-0.540 + 0.936i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (4.34 - 2.50i)T + (33.5 - 58.0i)T^{2}$$
71 $$1 + (-2.35 + 1.35i)T + (35.5 - 61.4i)T^{2}$$
73 $$1 + (-6.64 - 3.83i)T + (36.5 + 63.2i)T^{2}$$
79 $$1 + (-7.86 - 13.6i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 - 7.97iT - 83T^{2}$$
89 $$1 + 16.0iT - 89T^{2}$$
97 $$1 + (-12.3 + 7.11i)T + (48.5 - 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$