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Properties

Label 2-637-91.9-c1-0-17
Degree $2$
Conductor $637$
Sign $-0.429 - 0.902i$
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  + (0.866 + 1.5i)2-s + 0.732·3-s + (−0.5 + 0.866i)4-s + (−0.866 + 1.5i)5-s + (0.633 + 1.09i)6-s + 1.73·8-s − 2.46·9-s − 3·10-s + 4.73·11-s + (−0.366 + 0.633i)12-s + (1.59 + 3.23i)13-s + (−0.633 + 1.09i)15-s + (2.49 + 4.33i)16-s + (−2.13 + 3.69i)17-s + (−2.13 − 3.69i)18-s − 2·19-s + ⋯
L(s)  = 1  + (0.612 + 1.06i)2-s + 0.422·3-s + (−0.250 + 0.433i)4-s + (−0.387 + 0.670i)5-s + (0.258 + 0.448i)6-s + 0.612·8-s − 0.821·9-s − 0.948·10-s + 1.42·11-s + (−0.105 + 0.183i)12-s + (0.443 + 0.896i)13-s + (−0.163 + 0.283i)15-s + (0.624 + 1.08i)16-s + (−0.517 + 0.896i)17-s + (−0.502 − 0.871i)18-s − 0.458·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.429 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.429 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.429 - 0.902i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.429 - 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23962 + 1.96343i\)
\(L(\frac12)\) \(\approx\) \(1.23962 + 1.96343i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.59 - 3.23i)T \)
good2 \( 1 + (-0.866 - 1.5i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 - 0.732T + 3T^{2} \)
5 \( 1 + (0.866 - 1.5i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 4.73T + 11T^{2} \)
17 \( 1 + (2.13 - 3.69i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + (0.633 + 1.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.09 + 5.36i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.59 + 4.5i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.09 + 8.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.464 - 0.803i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.96 + 3.40i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.36 + 9.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 15.1T + 61T^{2} \)
67 \( 1 - 4.19T + 67T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.59 - 6.23i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.90 - 5.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.19T + 83T^{2} \)
89 \( 1 + (0.464 + 0.803i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.19 + 12.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.17527416462862618983369514717, −9.882395738654905314212254609162, −8.759841198423235984364590404310, −8.191530038452475614247062148685, −6.95412719244024001048982664383, −6.53310914041835502023902267148, −5.67350714520644949779872395030, −4.23342338249758058683365163570, −3.66016950619840959297246859945, −1.97547233017354786417857604930, 1.10538539451882305493523554086, 2.58382009843748109378909568535, 3.53948702228426859352325265408, 4.37522102676400650828390087537, 5.39533433714218930617399785973, 6.68107704362058089525126328432, 7.895448828451320914741161092056, 8.698202162408316214972797496022, 9.415309747765656711330748876679, 10.58650984723701051958539482836

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