# Properties

 Label 2-637-91.23-c1-0-18 Degree $2$ Conductor $637$ Sign $0.981 - 0.190i$ Analytic cond. $5.08647$ Root an. cond. $2.25532$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 2.10i·2-s + (−1.13 + 1.95i)3-s − 2.44·4-s + (−3.11 − 1.80i)5-s + (−4.13 − 2.38i)6-s − 0.948i·8-s + (−1.05 − 1.83i)9-s + (3.79 − 6.57i)10-s + (−0.767 − 0.443i)11-s + (2.76 − 4.79i)12-s + (−1.17 − 3.40i)13-s + (7.05 − 4.07i)15-s − 2.89·16-s + 4.96·17-s + (3.86 − 2.23i)18-s + (−2.06 + 1.18i)19-s + ⋯
 L(s)  = 1 + 1.49i·2-s + (−0.652 + 1.13i)3-s − 1.22·4-s + (−1.39 − 0.805i)5-s + (−1.68 − 0.973i)6-s − 0.335i·8-s + (−0.352 − 0.610i)9-s + (1.20 − 2.08i)10-s + (−0.231 − 0.133i)11-s + (0.799 − 1.38i)12-s + (−0.325 − 0.945i)13-s + (1.82 − 1.05i)15-s − 0.724·16-s + 1.20·17-s + (0.910 − 0.525i)18-s + (−0.472 + 0.272i)19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$637$$    =    $$7^{2} \cdot 13$$ Sign: $0.981 - 0.190i$ Analytic conductor: $$5.08647$$ Root analytic conductor: $$2.25532$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{637} (569, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 637,\ (\ :1/2),\ 0.981 - 0.190i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.279877 + 0.0269413i$$ $$L(\frac12)$$ $$\approx$$ $$0.279877 + 0.0269413i$$ $$L(\frac{3}{2})$$ 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 + (1.17 + 3.40i)T$$
good2 $$1 - 2.10iT - 2T^{2}$$
3 $$1 + (1.13 - 1.95i)T + (-1.5 - 2.59i)T^{2}$$
5 $$1 + (3.11 + 1.80i)T + (2.5 + 4.33i)T^{2}$$
11 $$1 + (0.767 + 0.443i)T + (5.5 + 9.52i)T^{2}$$
17 $$1 - 4.96T + 17T^{2}$$
19 $$1 + (2.06 - 1.18i)T + (9.5 - 16.4i)T^{2}$$
23 $$1 - 3.85T + 23T^{2}$$
29 $$1 + (0.640 + 1.11i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + (7.33 - 4.23i)T + (15.5 - 26.8i)T^{2}$$
37 $$1 + 9.63iT - 37T^{2}$$
41 $$1 + (-10.4 + 6.04i)T + (20.5 - 35.5i)T^{2}$$
43 $$1 + (1.82 - 3.15i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + (2.58 + 1.49i)T + (23.5 + 40.7i)T^{2}$$
53 $$1 + (2.46 + 4.26i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + 7.32iT - 59T^{2}$$
61 $$1 + (-0.769 - 1.33i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (7.29 + 4.21i)T + (33.5 + 58.0i)T^{2}$$
71 $$1 + (5.58 + 3.22i)T + (35.5 + 61.4i)T^{2}$$
73 $$1 + (6.19 - 3.57i)T + (36.5 - 63.2i)T^{2}$$
79 $$1 + (0.378 - 0.656i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 - 4.76iT - 83T^{2}$$
89 $$1 - 3.61iT - 89T^{2}$$
97 $$1 + (0.401 + 0.231i)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

−10.66218457837869239617400880671, −9.486734899815656767525801615451, −8.655461952623428258923987961015, −7.79863336778674014158904943483, −7.30896504159599332037273795504, −5.74909770472572142995237314045, −5.26485873916372720641514236346, −4.46062473389514182907802614410, −3.53927000098812710291553961415, −0.18629716494698232314646706253, 1.26170640179865343641330087164, 2.62558530090715252721349774550, 3.62895304599972484950548247093, 4.64381351064075529309195608240, 6.23122499516155403846394425987, 7.20794524287719515340080685423, 7.64416714546595045298096139132, 8.989940940502860085989094251871, 10.08062508167888862525208303079, 10.98921335374442295179669099631