Properties

Label 2-637-49.37-c1-0-55
Degree $2$
Conductor $637$
Sign $-0.989 - 0.144i$
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  + (2.30 − 1.56i)2-s + (−0.652 + 0.605i)3-s + (2.10 − 5.36i)4-s + (−3.46 − 1.06i)5-s + (−0.552 + 2.41i)6-s + (−2.11 + 1.58i)7-s + (−2.33 − 10.2i)8-s + (−0.164 + 2.20i)9-s + (−9.64 + 2.97i)10-s + (−0.296 − 3.96i)11-s + (1.87 + 4.77i)12-s + (−0.900 − 0.433i)13-s + (−2.38 + 6.97i)14-s + (2.90 − 1.39i)15-s + (−12.9 − 12.0i)16-s + (−0.592 − 0.0893i)17-s + ⋯
L(s)  = 1  + (1.62 − 1.10i)2-s + (−0.376 + 0.349i)3-s + (1.05 − 2.68i)4-s + (−1.54 − 0.477i)5-s + (−0.225 + 0.987i)6-s + (−0.800 + 0.599i)7-s + (−0.825 − 3.61i)8-s + (−0.0549 + 0.733i)9-s + (−3.05 + 0.940i)10-s + (−0.0894 − 1.19i)11-s + (0.541 + 1.37i)12-s + (−0.249 − 0.120i)13-s + (−0.636 + 1.86i)14-s + (0.750 − 0.361i)15-s + (−3.24 − 3.01i)16-s + (−0.143 − 0.0216i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.144i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.989 - 0.144i$
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: \((2,\ 637,\ (\ :1/2),\ -0.989 - 0.144i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.122556 + 1.68545i\)
\(L(\frac12)\) \(\approx\) \(0.122556 + 1.68545i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.11 - 1.58i)T \)
13 \( 1 + (0.900 + 0.433i)T \)
good2 \( 1 + (-2.30 + 1.56i)T + (0.730 - 1.86i)T^{2} \)
3 \( 1 + (0.652 - 0.605i)T + (0.224 - 2.99i)T^{2} \)
5 \( 1 + (3.46 + 1.06i)T + (4.13 + 2.81i)T^{2} \)
11 \( 1 + (0.296 + 3.96i)T + (-10.8 + 1.63i)T^{2} \)
17 \( 1 + (0.592 + 0.0893i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (-2.14 + 3.72i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.46 + 0.521i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (-1.94 + 2.44i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (0.354 + 0.614i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.299 + 0.762i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (-1.60 - 7.04i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (0.644 - 2.82i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-7.11 + 4.85i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (-0.911 + 2.32i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (-7.00 + 2.15i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (1.62 + 4.13i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (6.97 + 12.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-8.29 - 10.3i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (9.49 + 6.47i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 + (8.59 - 14.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.85 - 2.33i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-1.00 + 13.3i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 - 8.83T + 97T^{2} \)
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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

−10.65484679183474573652635377665, −9.632600968893269471264994988553, −8.528512087481680959742908070052, −7.15373397729851911030121423748, −5.99934086854671371646210826325, −5.15879556597491400051937933661, −4.45132081056487296198861702151, −3.45168860035221009834941767015, −2.68842424012513717106997289923, −0.55083298547130682979290769665, 2.97986041759664856597309093990, 3.81070857858875310211483824150, 4.43187970161294264058194042616, 5.68117579588369655503977535585, 6.75556560095986479617229966277, 7.19842804958286446916128858777, 7.63761071592623383921069340370, 8.926221304855437700088006928359, 10.52174923558507096420678067088, 11.60806686263246423046968204671

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