Properties

Label 2-637-91.59-c1-0-10
Degree $2$
Conductor $637$
Sign $0.995 + 0.0982i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.18 − 1.18i)2-s + (0.552 − 0.318i)3-s + 0.809i·4-s + (−1.94 − 0.520i)5-s + (−1.03 − 0.276i)6-s + (−1.41 + 1.41i)8-s + (−1.29 + 2.24i)9-s + (1.68 + 2.91i)10-s + (−0.948 − 0.254i)11-s + (0.258 + 0.446i)12-s + (1.60 + 3.22i)13-s + (−1.23 + 0.331i)15-s + 4.96·16-s + 5.98·17-s + (4.19 − 1.12i)18-s + (0.726 + 2.71i)19-s + ⋯
L(s)  = 1  + (−0.838 − 0.838i)2-s + (0.318 − 0.184i)3-s + 0.404i·4-s + (−0.868 − 0.232i)5-s + (−0.421 − 0.112i)6-s + (−0.498 + 0.498i)8-s + (−0.432 + 0.748i)9-s + (0.532 + 0.922i)10-s + (−0.285 − 0.0766i)11-s + (0.0744 + 0.129i)12-s + (0.446 + 0.894i)13-s + (−0.319 + 0.0856i)15-s + 1.24·16-s + 1.45·17-s + (0.989 − 0.265i)18-s + (0.166 + 0.622i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.995 + 0.0982i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.995 + 0.0982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.713763 - 0.0351334i\)
\(L(\frac12)\) \(\approx\) \(0.713763 - 0.0351334i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.60 - 3.22i)T \)
good2 \( 1 + (1.18 + 1.18i)T + 2iT^{2} \)
3 \( 1 + (-0.552 + 0.318i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.94 + 0.520i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.948 + 0.254i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 - 5.98T + 17T^{2} \)
19 \( 1 + (-0.726 - 2.71i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 2.98iT - 23T^{2} \)
29 \( 1 + (-3.65 + 6.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.23 - 8.34i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (3.39 - 3.39i)T - 37iT^{2} \)
41 \( 1 + (-0.886 - 3.30i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (0.748 - 0.432i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.794 + 2.96i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.16 - 5.47i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.359 - 0.359i)T + 59iT^{2} \)
61 \( 1 + (-11.2 - 6.51i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.221 + 0.827i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (3.01 - 11.2i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (1.40 - 0.377i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.80 - 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.23 + 1.23i)T - 83iT^{2} \)
89 \( 1 + (5.67 + 5.67i)T + 89iT^{2} \)
97 \( 1 + (-12.0 - 3.23i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.42020971235078067320656638644, −9.908445410546681798341935893750, −8.616413151492541140182267881647, −8.330041101946419065681402596102, −7.47700421931175036470366442013, −6.06680484371880923756750602545, −4.94173385755474189528677323201, −3.60340444658368558162374376072, −2.52773460287174272471135043245, −1.20706068642651251545832153647, 0.58069891695821916258969007881, 3.14533246522692643076968967908, 3.68642725644974691687326264841, 5.38616893266321006500408656100, 6.32382185273409387981107166052, 7.42900572198446193350530195912, 7.898625464652720683541711098712, 8.677541696544061436437275668664, 9.505604128015299108113014046261, 10.29268674226072336150361126355

Graph of the $Z$-function along the critical line