# Properties

 Label 2-637-91.4-c1-0-35 Degree $2$ Conductor $637$ Sign $-0.617 + 0.786i$ Analytic cond. $5.08647$ Root an. cond. $2.25532$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $1$

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## Dirichlet series

 L(s)  = 1 + 1.73i·2-s + (−1 − 1.73i)3-s − 0.999·4-s + (−1.5 + 0.866i)5-s + (2.99 − 1.73i)6-s + 1.73i·8-s + (−0.499 + 0.866i)9-s + (−1.49 − 2.59i)10-s + (0.999 + 1.73i)12-s + (−2.5 − 2.59i)13-s + (3 + 1.73i)15-s − 5·16-s − 3·17-s + (−1.49 − 0.866i)18-s + (3 + 1.73i)19-s + (1.49 − 0.866i)20-s + ⋯
 L(s)  = 1 + 1.22i·2-s + (−0.577 − 0.999i)3-s − 0.499·4-s + (−0.670 + 0.387i)5-s + (1.22 − 0.707i)6-s + 0.612i·8-s + (−0.166 + 0.288i)9-s + (−0.474 − 0.821i)10-s + (0.288 + 0.499i)12-s + (−0.693 − 0.720i)13-s + (0.774 + 0.447i)15-s − 1.25·16-s − 0.727·17-s + (−0.353 − 0.204i)18-s + (0.688 + 0.397i)19-s + (0.335 − 0.193i)20-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 + 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$637$$    =    $$7^{2} \cdot 13$$ Sign: $-0.617 + 0.786i$ 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: $$1$$ Selberg data: $$(2,\ 637,\ (\ :1/2),\ -0.617 + 0.786i)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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 - 1.73iT - 2T^{2}$$
3 $$1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2}$$
5 $$1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2}$$
11 $$1 + (5.5 - 9.52i)T^{2}$$
17 $$1 + 3T + 17T^{2}$$
19 $$1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2}$$
23 $$1 + 6T + 23T^{2}$$
29 $$1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + (3 + 1.73i)T + (15.5 + 26.8i)T^{2}$$
37 $$1 + 8.66iT - 37T^{2}$$
41 $$1 + (4.5 + 2.59i)T + (20.5 + 35.5i)T^{2}$$
43 $$1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + (3 - 1.73i)T + (23.5 - 40.7i)T^{2}$$
53 $$1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 - 6.92iT - 59T^{2}$$
61 $$1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (3 - 1.73i)T + (33.5 - 58.0i)T^{2}$$
71 $$1 + (-3 + 1.73i)T + (35.5 - 61.4i)T^{2}$$
73 $$1 + (1.5 + 0.866i)T + (36.5 + 63.2i)T^{2}$$
79 $$1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 - 13.8iT - 83T^{2}$$
89 $$1 - 6.92iT - 89T^{2}$$
97 $$1 + (-6 + 3.46i)T + (48.5 - 84.0i)T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−10.35205602390525889272931606696, −9.088644749570592181754073860565, −7.945874074688101266026594056795, −7.44893837127780027863693283752, −6.88285709162521628742163938014, −5.93535167488051669593184991727, −5.24554852569514550660462436270, −3.75715249035488210127176405351, −2.09535498099824651800542608901, 0, 1.92348803860528736841856208150, 3.34938697862149246183232929567, 4.36391823464928821930099339231, 4.81258252093320177055724164477, 6.28380140671001822267107925153, 7.42644763046092665185879553436, 8.602594549874780978122679530175, 9.777222804874365451493773048998, 9.941604923339914191669973967770