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Properties

Label 2-637-91.88-c1-0-14
Degree $2$
Conductor $637$
Sign $0.794 + 0.606i$
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  + (−1.89 − 1.09i)2-s − 1.79·3-s + (1.39 + 2.41i)4-s + (1.89 − 1.09i)5-s + (3.39 + 1.96i)6-s − 1.73i·8-s + 0.208·9-s − 4.79·10-s − 1.27i·11-s + (−2.5 − 4.33i)12-s + (3.5 + 0.866i)13-s + (−3.39 + 1.96i)15-s + (0.895 − 1.55i)16-s + (−1.5 − 2.59i)17-s + (−0.395 − 0.228i)18-s + 6.56i·19-s + ⋯
L(s)  = 1  + (−1.34 − 0.773i)2-s − 1.03·3-s + (0.697 + 1.20i)4-s + (0.847 − 0.489i)5-s + (1.38 + 0.800i)6-s − 0.612i·8-s + 0.0695·9-s − 1.51·10-s − 0.384i·11-s + (−0.721 − 1.24i)12-s + (0.970 + 0.240i)13-s + (−0.876 + 0.506i)15-s + (0.223 − 0.387i)16-s + (−0.363 − 0.630i)17-s + (−0.0932 − 0.0538i)18-s + 1.50i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.794 + 0.606i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.794 + 0.606i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.526759 - 0.178146i\)
\(L(\frac12)\) \(\approx\) \(0.526759 - 0.178146i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.5 - 0.866i)T \)
good2 \( 1 + (1.89 + 1.09i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + 1.79T + 3T^{2} \)
5 \( 1 + (-1.89 + 1.09i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 1.27iT - 11T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 6.56iT - 19T^{2} \)
23 \( 1 + (3.79 - 6.56i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.10 - 1.91i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-7.5 - 4.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (6 + 3.46i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.20 + 1.27i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.18 + 3.78i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.70 + 2.14i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.08 + 10.5i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.66 + 4.42i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 12.7T + 61T^{2} \)
67 \( 1 - 11.4iT - 67T^{2} \)
71 \( 1 + (0.791 + 0.456i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (3 + 1.73i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.55iT - 83T^{2} \)
89 \( 1 + (-2.52 - 1.45i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.1 + 7.61i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.31191675199481284895098516933, −9.891176901252552231379280499662, −8.824339278421192262467053083100, −8.341529942917155286840608546271, −7.01797988008727037178015339679, −5.83910227222597428154607054059, −5.33606203996878601312637995115, −3.59456460822967411823273910689, −1.99830548317027371002894820121, −0.941230193954480565229120074302, 0.74887727119996927461514584847, 2.42799224242947469819032395164, 4.44343398440299428151454173163, 5.84295557175905360746108664247, 6.31290557489062687614342156901, 6.90956719701582355528530522475, 8.191953121935581103332238307055, 8.834917344490199436722824344798, 9.887592606916230005494386425965, 10.49207969200666037937496824950

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