Properties

Label 2-637-49.4-c1-0-31
Degree $2$
Conductor $637$
Sign $0.0809 + 0.996i$
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  + (−0.434 − 0.295i)2-s + (0.0114 + 0.0106i)3-s + (−0.629 − 1.60i)4-s + (3.15 − 0.973i)5-s + (−0.00182 − 0.00800i)6-s + (0.290 + 2.62i)7-s + (−0.435 + 1.90i)8-s + (−0.224 − 2.99i)9-s + (−1.65 − 0.511i)10-s + (0.374 − 4.99i)11-s + (0.00983 − 0.0250i)12-s + (0.900 − 0.433i)13-s + (0.652 − 1.22i)14-s + (0.0464 + 0.0223i)15-s + (−1.77 + 1.64i)16-s + (2.27 − 0.342i)17-s + ⋯
L(s)  = 1  + (−0.306 − 0.209i)2-s + (0.00661 + 0.00613i)3-s + (−0.314 − 0.802i)4-s + (1.41 − 0.435i)5-s + (−0.000745 − 0.00326i)6-s + (0.109 + 0.993i)7-s + (−0.153 + 0.674i)8-s + (−0.0747 − 0.997i)9-s + (−0.524 − 0.161i)10-s + (0.112 − 1.50i)11-s + (0.00284 − 0.00723i)12-s + (0.249 − 0.120i)13-s + (0.174 − 0.328i)14-s + (0.0120 + 0.00577i)15-s + (−0.443 + 0.411i)16-s + (0.550 − 0.0829i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0809 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0809 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.0809 + 0.996i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.0809 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08729 - 1.00259i\)
\(L(\frac12)\) \(\approx\) \(1.08729 - 1.00259i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.290 - 2.62i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
good2 \( 1 + (0.434 + 0.295i)T + (0.730 + 1.86i)T^{2} \)
3 \( 1 + (-0.0114 - 0.0106i)T + (0.224 + 2.99i)T^{2} \)
5 \( 1 + (-3.15 + 0.973i)T + (4.13 - 2.81i)T^{2} \)
11 \( 1 + (-0.374 + 4.99i)T + (-10.8 - 1.63i)T^{2} \)
17 \( 1 + (-2.27 + 0.342i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (-3.22 - 5.57i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.99 + 0.451i)T + (21.9 + 6.77i)T^{2} \)
29 \( 1 + (3.36 + 4.21i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + (0.250 - 0.433i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.24 + 8.26i)T + (-27.1 - 25.1i)T^{2} \)
41 \( 1 + (1.00 - 4.38i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (0.292 + 1.27i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (3.48 + 2.37i)T + (17.1 + 43.7i)T^{2} \)
53 \( 1 + (4.37 + 11.1i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (-12.3 - 3.82i)T + (48.7 + 33.2i)T^{2} \)
61 \( 1 + (1.78 - 4.54i)T + (-44.7 - 41.4i)T^{2} \)
67 \( 1 + (-0.633 + 1.09i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.72 - 2.16i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-2.41 + 1.64i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (-0.610 - 1.05i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.91 - 3.81i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-0.387 - 5.17i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 + 5.70T + 97T^{2} \)
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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

−9.986594797278378079810575760419, −9.615072742507023179112798510544, −8.858083303031200268053960314291, −8.200162744953209252091052182336, −6.21245430646758226103838132103, −5.84418513611389398741428306046, −5.33085217016301144193177886007, −3.55755959473779876147138845867, −2.12979444694040607107105710157, −0.977197729174345447694341169271, 1.71855740964455899684569088232, 2.95932327322804077241857538394, 4.36477068858579020655244964750, 5.21986108479109099858492322801, 6.63289565402843813654293689453, 7.27332416888619383868793499210, 7.987692751106128433354970427234, 9.274999811128491512819343952996, 9.829435678718424039586523654903, 10.50813842180679280621595073967

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