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Properties

Label 2-637-91.88-c1-0-41
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.395 + 0.228i)2-s − 2.79·3-s + (−0.895 − 1.55i)4-s + (0.395 − 0.228i)5-s + (−1.10 − 0.637i)6-s − 1.73i·8-s + 4.79·9-s + 0.208·10-s − 3.92i·11-s + (2.5 + 4.33i)12-s + (−3.5 − 0.866i)13-s + (−1.10 + 0.637i)15-s + (−1.39 + 2.41i)16-s + (1.5 + 2.59i)17-s + (1.89 + 1.09i)18-s + 1.37i·19-s + ⋯
L(s)  = 1  + (0.279 + 0.161i)2-s − 1.61·3-s + (−0.447 − 0.775i)4-s + (0.176 − 0.102i)5-s + (−0.450 − 0.260i)6-s − 0.612i·8-s + 1.59·9-s + 0.0660·10-s − 1.18i·11-s + (0.721 + 1.24i)12-s + (−0.970 − 0.240i)13-s + (−0.285 + 0.164i)15-s + (−0.348 + 0.604i)16-s + (0.363 + 0.630i)17-s + (0.446 + 0.257i)18-s + 0.314i·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: \(1\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.794 - 0.606i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.395 - 0.228i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + 2.79T + 3T^{2} \)
5 \( 1 + (-0.395 + 0.228i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 3.92iT - 11T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 1.37iT - 19T^{2} \)
23 \( 1 + (-0.791 + 1.37i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.39 - 5.88i)T + (-14.5 + 25.1i)T^{2} \)
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 \( 1 + (6.79 - 3.92i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.68 - 8.11i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.29 - 4.78i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.08 - 5.33i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-10.6 + 6.15i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 14.7T + 61T^{2} \)
67 \( 1 + 4.47iT - 67T^{2} \)
71 \( 1 + (-3.79 - 2.18i)T + (35.5 + 61.4i)T^{2} \)
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 \( 1 + 7.02iT - 83T^{2} \)
89 \( 1 + (13.9 + 8.07i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.31 - 3.64i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.20743402819269398667772228648, −9.514966218346523668108605171196, −8.303402142257381408112723300160, −6.95645538376390165431511026074, −6.17124881041506771497473506334, −5.40875844360904347246112804037, −4.98262095179010938913852051827, −3.63688282479232588270682973216, −1.38228298275911195473528000535, 0, 2.19183216262703043005804863449, 3.83316924665352311824692119410, 5.00027443910550228092071050657, 5.21519282429262463344276603999, 6.80135741074868912472326485978, 7.18905558440256622046945218427, 8.456586967780593685694132406645, 9.768161421122547893614309885470, 10.18236098644826566047376551066

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