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Properties

Label 2-637-91.81-c1-0-19
Degree $2$
Conductor $637$
Sign $0.191 - 0.981i$
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  + (−0.866 + 1.5i)2-s + 2.73·3-s + (−0.5 − 0.866i)4-s + (−0.866 − 1.5i)5-s + (−2.36 + 4.09i)6-s − 1.73·8-s + 4.46·9-s + 3·10-s + 1.26·11-s + (−1.36 − 2.36i)12-s + (3.59 + 0.232i)13-s + (−2.36 − 4.09i)15-s + (2.49 − 4.33i)16-s + (3.86 + 6.69i)17-s + (−3.86 + 6.69i)18-s + 2·19-s + ⋯
L(s)  = 1  + (−0.612 + 1.06i)2-s + 1.57·3-s + (−0.250 − 0.433i)4-s + (−0.387 − 0.670i)5-s + (−0.965 + 1.67i)6-s − 0.612·8-s + 1.48·9-s + 0.948·10-s + 0.382·11-s + (−0.394 − 0.683i)12-s + (0.997 + 0.0643i)13-s + (−0.610 − 1.05i)15-s + (0.624 − 1.08i)16-s + (0.937 + 1.62i)17-s + (−0.911 + 1.57i)18-s + 0.458·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.191 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.191 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.191 - 0.981i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.191 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40759 + 1.15908i\)
\(L(\frac12)\) \(\approx\) \(1.40759 + 1.15908i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.59 - 0.232i)T \)
good2 \( 1 + (0.866 - 1.5i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 - 2.73T + 3T^{2} \)
5 \( 1 + (0.866 + 1.5i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 1.26T + 11T^{2} \)
17 \( 1 + (-3.86 - 6.69i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (2.36 - 4.09i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.09 - 3.63i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.59 - 4.5i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.0980 + 0.169i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.46 + 11.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.96 + 8.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.63 + 6.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 4.80T + 61T^{2} \)
67 \( 1 + 6.19T + 67T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.59 + 2.76i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.09 + 14.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.19T + 83T^{2} \)
89 \( 1 + (6.46 - 11.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.19 - 5.53i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.31402630109114105858845859878, −9.372829074900636286735089762741, −8.725632452493697133955993939082, −8.188374080144534919463947613509, −7.68789135010208506338016463596, −6.59689423671353032571441097843, −5.55395417790959195666589958763, −3.94774941028514516189247652443, −3.27751636068378832141753391100, −1.51588341035377775840233055427, 1.26131956013346689223843020942, 2.72888216077880689284527216888, 3.11465916460570657844776950902, 4.16689541323760932524433080054, 6.01442183063829316089581830239, 7.26697016185631552067609119228, 8.013595192562076271009831623364, 8.914329620989518050516511631241, 9.506868130332146647465106590236, 10.20990847117871567234668439248

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