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Properties

Label 2-637-91.31-c1-0-34
Degree $2$
Conductor $637$
Sign $0.147 + 0.989i$
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  + (0.550 + 0.147i)2-s + (1.06 + 0.616i)3-s + (−1.45 − 0.837i)4-s + (0.377 − 1.41i)5-s + (0.496 + 0.496i)6-s + (−1.48 − 1.48i)8-s + (−0.740 − 1.28i)9-s + (0.416 − 0.720i)10-s + (0.815 − 0.218i)11-s + (−1.03 − 1.78i)12-s + (−3.59 − 0.296i)13-s + (1.27 − 1.27i)15-s + (1.07 + 1.86i)16-s + (3.67 − 6.36i)17-s + (−0.218 − 0.815i)18-s + (1.31 − 4.90i)19-s + ⋯
L(s)  = 1  + (0.389 + 0.104i)2-s + (0.616 + 0.355i)3-s + (−0.725 − 0.418i)4-s + (0.168 − 0.630i)5-s + (0.202 + 0.202i)6-s + (−0.523 − 0.523i)8-s + (−0.246 − 0.427i)9-s + (0.131 − 0.227i)10-s + (0.245 − 0.0658i)11-s + (−0.297 − 0.516i)12-s + (−0.996 − 0.0822i)13-s + (0.328 − 0.328i)15-s + (0.269 + 0.466i)16-s + (0.891 − 1.54i)17-s + (−0.0515 − 0.192i)18-s + (0.301 − 1.12i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.147 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.147 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.147 + 0.989i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.147 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19338 - 1.02834i\)
\(L(\frac12)\) \(\approx\) \(1.19338 - 1.02834i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.59 + 0.296i)T \)
good2 \( 1 + (-0.550 - 0.147i)T + (1.73 + i)T^{2} \)
3 \( 1 + (-1.06 - 0.616i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.377 + 1.41i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.815 + 0.218i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-3.67 + 6.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.31 + 4.90i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.84 - 2.22i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.54T + 29T^{2} \)
31 \( 1 + (-1.73 + 0.465i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.05 + 3.93i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.23 - 1.23i)T + 41iT^{2} \)
43 \( 1 - 8.66iT - 43T^{2} \)
47 \( 1 + (3.44 + 0.923i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-4.89 + 8.48i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.398 + 1.48i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-6.15 + 3.55i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.20 - 11.9i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.46 - 1.46i)T - 71iT^{2} \)
73 \( 1 + (0.0380 + 0.141i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.39 - 4.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-12.3 - 12.3i)T + 83iT^{2} \)
89 \( 1 + (-9.41 - 2.52i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (6.05 + 6.05i)T + 97iT^{2} \)
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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

−9.811690904663680374387036666007, −9.579270480340234362284387208556, −8.927918722155069124740458530207, −7.892447739841957582022057879982, −6.74861628095074938650567555077, −5.45980754045449858541997568687, −4.92238864287000527797730670802, −3.83924800891921852092987963027, −2.75467238548261556264421382380, −0.72375627899388536199213264804, 2.02594560692865242653338507930, 3.13143574881461538473723001793, 4.02372202498973870564028993068, 5.25536698616122413052309751631, 6.20343467844888015172831151354, 7.53367622785925421718627956020, 8.068303112209287731988209488029, 8.896153183401311291070500197645, 9.978175403434531269360052321688, 10.60128723190865913425754808271

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