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Properties

Label 2-637-7.4-c1-0-37
Degree $2$
Conductor $637$
Sign $-0.605 - 0.795i$
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  + (−0.707 − 1.22i)2-s + (0.707 − 1.22i)3-s + (−2.20 − 3.82i)5-s − 2·6-s − 2.82·8-s + (0.500 + 0.866i)9-s + (−3.12 + 5.40i)10-s + (2.12 − 3.67i)11-s − 13-s − 6.24·15-s + (2.00 + 3.46i)16-s + (0.707 − 1.22i)17-s + (0.707 − 1.22i)18-s + (−0.621 − 1.07i)19-s − 6·22-s + (0.0857 + 0.148i)23-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)2-s + (0.408 − 0.707i)3-s + (−0.987 − 1.70i)5-s − 0.816·6-s − 0.999·8-s + (0.166 + 0.288i)9-s + (−0.987 + 1.70i)10-s + (0.639 − 1.10i)11-s − 0.277·13-s − 1.61·15-s + (0.500 + 0.866i)16-s + (0.171 − 0.297i)17-s + (0.166 − 0.288i)18-s + (−0.142 − 0.246i)19-s − 1.27·22-s + (0.0178 + 0.0309i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.435477 + 0.878461i\)
\(L(\frac12)\) \(\approx\) \(0.435477 + 0.878461i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good2 \( 1 + (0.707 + 1.22i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.707 + 1.22i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.20 + 3.82i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.12 + 3.67i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.707 + 1.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.621 + 1.07i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.0857 - 0.148i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.82T + 29T^{2} \)
31 \( 1 + (-2.62 + 4.54i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.12 - 5.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.17T + 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (2.20 + 3.82i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.91 + 5.04i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.82 - 10.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.24 - 2.15i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.07T + 71T^{2} \)
73 \( 1 + (-0.378 + 0.655i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.742 - 1.28i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.75T + 83T^{2} \)
89 \( 1 + (2.20 + 3.82i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 13.7T + 97T^{2} \)
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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

−9.911611071121596212563920079693, −9.013031209766180282432741745332, −8.430769288923763980473233015533, −7.82033781167227163561025773279, −6.52703594860114808257873580176, −5.28555950376612400791514332397, −4.21938404747790318117721286375, −2.94196093513247449332513168048, −1.48219492623847272883945898622, −0.63924813676598929352401460753, 2.70969450347938496056195857568, 3.57050807769704823976783886776, 4.43785922933592142099961430160, 6.31221564393132257067580687216, 6.83678929496537229541908280635, 7.55185644320047912807710415014, 8.330664170087626497799734612701, 9.405894285816465110330029308166, 10.10564897225060019790625778978, 10.90560348233766062819855495140

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