# Properties

 Label 2-637-91.24-c1-0-30 Degree $2$ Conductor $637$ Sign $-0.747 + 0.664i$ 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 + (−0.813 + 0.218i)2-s − 1.43i·3-s + (−1.11 + 0.645i)4-s + (−1.38 − 0.370i)5-s + (0.312 + 1.16i)6-s + (1.95 − 1.95i)8-s + 0.945·9-s + 1.20·10-s + (3.43 − 3.43i)11-s + (0.924 + 1.60i)12-s + (−1.04 + 3.44i)13-s + (−0.531 + 1.98i)15-s + (0.122 − 0.212i)16-s + (−1.49 − 2.58i)17-s + (−0.769 + 0.206i)18-s + (−4.44 + 4.44i)19-s + ⋯
 L(s)  = 1 + (−0.575 + 0.154i)2-s − 0.827i·3-s + (−0.558 + 0.322i)4-s + (−0.618 − 0.165i)5-s + (0.127 + 0.476i)6-s + (0.692 − 0.692i)8-s + 0.315·9-s + 0.381·10-s + (1.03 − 1.03i)11-s + (0.266 + 0.462i)12-s + (−0.291 + 0.956i)13-s + (−0.137 + 0.512i)15-s + (0.0306 − 0.0531i)16-s + (−0.361 − 0.626i)17-s + (−0.181 + 0.0486i)18-s + (−1.01 + 1.01i)19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.747 + 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$637$$    =    $$7^{2} \cdot 13$$ Sign: $-0.747 + 0.664i$ Analytic conductor: $$5.08647$$ Root analytic conductor: $$2.25532$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{637} (570, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 637,\ (\ :1/2),\ -0.747 + 0.664i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.183820 - 0.483218i$$ $$L(\frac12)$$ $$\approx$$ $$0.183820 - 0.483218i$$ $$L(\frac{3}{2})$$ 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 + (1.04 - 3.44i)T$$
good2 $$1 + (0.813 - 0.218i)T + (1.73 - i)T^{2}$$
3 $$1 + 1.43iT - 3T^{2}$$
5 $$1 + (1.38 + 0.370i)T + (4.33 + 2.5i)T^{2}$$
11 $$1 + (-3.43 + 3.43i)T - 11iT^{2}$$
17 $$1 + (1.49 + 2.58i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (4.44 - 4.44i)T - 19iT^{2}$$
23 $$1 + (-1.02 - 0.590i)T + (11.5 + 19.9i)T^{2}$$
29 $$1 + (2.77 + 4.81i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + (1.75 + 6.54i)T + (-26.8 + 15.5i)T^{2}$$
37 $$1 + (1.44 + 5.40i)T + (-32.0 + 18.5i)T^{2}$$
41 $$1 + (4.71 + 1.26i)T + (35.5 + 20.5i)T^{2}$$
43 $$1 + (2.90 + 1.67i)T + (21.5 + 37.2i)T^{2}$$
47 $$1 + (-1.51 + 5.63i)T + (-40.7 - 23.5i)T^{2}$$
53 $$1 + (2.89 - 5.01i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (2.88 - 10.7i)T + (-51.0 - 29.5i)T^{2}$$
61 $$1 + 9.18iT - 61T^{2}$$
67 $$1 + (1.38 + 1.38i)T + 67iT^{2}$$
71 $$1 + (3.19 - 0.855i)T + (61.4 - 35.5i)T^{2}$$
73 $$1 + (0.482 - 0.129i)T + (63.2 - 36.5i)T^{2}$$
79 $$1 + (-3.47 - 6.02i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + (3.22 - 3.22i)T - 83iT^{2}$$
89 $$1 + (0.237 - 0.0636i)T + (77.0 - 44.5i)T^{2}$$
97 $$1 + (2.43 + 9.08i)T + (-84.0 + 48.5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$