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Properties

Label 2-637-91.45-c1-0-40
Degree $2$
Conductor $637$
Sign $-0.876 + 0.480i$
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  + (1.74 − 1.74i)2-s + (0.146 − 0.0844i)3-s − 4.08i·4-s + (0.638 − 2.38i)5-s + (0.107 − 0.402i)6-s + (−3.63 − 3.63i)8-s + (−1.48 + 2.57i)9-s + (−3.04 − 5.26i)10-s + (1.17 − 4.40i)11-s + (−0.344 − 0.596i)12-s + (−1.54 + 3.25i)13-s + (−0.107 − 0.402i)15-s − 4.49·16-s − 0.112·17-s + (1.89 + 7.07i)18-s + (3.32 − 0.891i)19-s + ⋯
L(s)  = 1  + (1.23 − 1.23i)2-s + (0.0844 − 0.0487i)3-s − 2.04i·4-s + (0.285 − 1.06i)5-s + (0.0440 − 0.164i)6-s + (−1.28 − 1.28i)8-s + (−0.495 + 0.857i)9-s + (−0.962 − 1.66i)10-s + (0.355 − 1.32i)11-s + (−0.0994 − 0.172i)12-s + (−0.429 + 0.903i)13-s + (−0.0278 − 0.103i)15-s − 1.12·16-s − 0.0273·17-s + (0.447 + 1.66i)18-s + (0.763 − 0.204i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.480i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.876 + 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.876 + 0.480i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.876 + 0.480i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.715347 - 2.79370i\)
\(L(\frac12)\) \(\approx\) \(0.715347 - 2.79370i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.54 - 3.25i)T \)
good2 \( 1 + (-1.74 + 1.74i)T - 2iT^{2} \)
3 \( 1 + (-0.146 + 0.0844i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.638 + 2.38i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.17 + 4.40i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + 0.112T + 17T^{2} \)
19 \( 1 + (-3.32 + 0.891i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 0.652iT - 23T^{2} \)
29 \( 1 + (2.82 - 4.88i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.34 - 1.43i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.09 - 1.09i)T + 37iT^{2} \)
41 \( 1 + (-10.6 + 2.86i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.08 + 3.51i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.72 - 1.53i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.41 - 4.17i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.79 - 2.79i)T - 59iT^{2} \)
61 \( 1 + (13.2 + 7.66i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.07 + 1.62i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.56 - 0.955i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.651 - 2.43i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.11 - 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.34 - 3.34i)T + 83iT^{2} \)
89 \( 1 + (-6.14 + 6.14i)T - 89iT^{2} \)
97 \( 1 + (1.04 - 3.89i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.76094486202397818222070442433, −9.303096556913863754454476490810, −8.946557334342431684066006876774, −7.57987875257374832212715804482, −6.00993020031572771238431166735, −5.33302281799546997244569163939, −4.58140783026077140024495757998, −3.49055919593423404713901848419, −2.35577282769543807971685560525, −1.16222981980571818620141814527, 2.62614064814842306756634520917, 3.56323771068603108954410169282, 4.57366949779653967219614637557, 5.77089732546000175101651531269, 6.26776706088267206111823210222, 7.38514157402292840240239858045, 7.60927940789041639440753115354, 9.161352593967600222176617965306, 9.961238983069923931021464767972, 11.11885980779537342993356641649

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