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Properties

Label 2-637-91.54-c1-0-15
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: $\chi_{637} (509, \cdot )$
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

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

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