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Properties

Label 2-637-49.37-c1-0-24
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: $\chi_{637} (625, \cdot )$
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

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

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