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Properties

Label 2-637-13.3-c1-0-16
Degree $2$
Conductor $637$
Sign $-0.999 + 0.0251i$
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  + (−1.21 − 2.10i)2-s + (−0.376 − 0.652i)3-s + (−1.95 + 3.39i)4-s + 0.341·5-s + (−0.916 + 1.58i)6-s + 4.65·8-s + (1.21 − 2.10i)9-s + (−0.415 − 0.719i)10-s + (1.21 + 2.10i)11-s + 2.95·12-s + (2.50 − 2.59i)13-s + (−0.128 − 0.222i)15-s + (−1.74 − 3.02i)16-s + (0.974 − 1.68i)17-s − 5.91·18-s + (3.14 − 5.44i)19-s + ⋯
L(s)  = 1  + (−0.859 − 1.48i)2-s + (−0.217 − 0.376i)3-s + (−0.978 + 1.69i)4-s + 0.152·5-s + (−0.374 + 0.647i)6-s + 1.64·8-s + (0.405 − 0.702i)9-s + (−0.131 − 0.227i)10-s + (0.366 + 0.635i)11-s + 0.851·12-s + (0.693 − 0.720i)13-s + (−0.0332 − 0.0575i)15-s + (−0.437 − 0.757i)16-s + (0.236 − 0.409i)17-s − 1.39·18-s + (0.721 − 1.24i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.999 + 0.0251i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.999 + 0.0251i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0102661 - 0.817542i\)
\(L(\frac12)\) \(\approx\) \(0.0102661 - 0.817542i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.50 + 2.59i)T \)
good2 \( 1 + (1.21 + 2.10i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.376 + 0.652i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 0.341T + 5T^{2} \)
11 \( 1 + (-1.21 - 2.10i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.974 + 1.68i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.14 + 5.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.84 - 3.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.22 + 3.84i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.97T + 31T^{2} \)
37 \( 1 + (-4.81 - 8.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.26 + 10.8i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.20 + 7.28i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.00T + 47T^{2} \)
53 \( 1 - 1.49T + 53T^{2} \)
59 \( 1 + (0.313 - 0.542i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.571 + 0.990i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.79 - 4.84i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.74 - 8.22i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 - 4.47T + 79T^{2} \)
83 \( 1 - 1.41T + 83T^{2} \)
89 \( 1 + (6.22 + 10.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.13 - 8.90i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.994076583383720241507836044818, −9.587215429734493864531662314111, −8.774109107701047404091499646453, −7.70159368046002348768788674314, −6.83605521799790520831705036074, −5.50841613685318383193933388313, −4.04034477824557599164036466264, −3.12516346707865008786577864964, −1.80515769542522829470395819796, −0.68533658273075388308636848416, 1.46984545366465835967030062283, 3.74731159138312938036909757254, 4.97158100281800306629645489461, 5.89582806615182201394609405300, 6.50675026437480524934834370217, 7.67504114297580053763741108412, 8.186273752422438405350178877444, 9.223482376379168575480517408679, 9.794143783776285042972786072993, 10.71340135218825846508186972471

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