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Properties

Label 2-637-13.10-c1-0-35
Degree $2$
Conductor $637$
Sign $0.980 + 0.195i$
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 − 1.33i)2-s + (1.59 + 2.76i)3-s + (2.54 − 4.40i)4-s + 0.329i·5-s + (7.35 + 4.24i)6-s − 8.21i·8-s + (−3.58 + 6.21i)9-s + (0.438 + 0.759i)10-s + (0.411 − 0.237i)11-s + 16.2·12-s + (−2.74 − 2.34i)13-s + (−0.909 + 0.525i)15-s + (−5.84 − 10.1i)16-s + (−0.626 + 1.08i)17-s + 19.0i·18-s + (−3.06 − 1.76i)19-s + ⋯
L(s)  = 1  + (1.62 − 0.941i)2-s + (0.920 + 1.59i)3-s + (1.27 − 2.20i)4-s + 0.147i·5-s + (3.00 + 1.73i)6-s − 2.90i·8-s + (−1.19 + 2.07i)9-s + (0.138 + 0.240i)10-s + (0.124 − 0.0716i)11-s + 4.68·12-s + (−0.760 − 0.649i)13-s + (−0.234 + 0.135i)15-s + (−1.46 − 2.52i)16-s + (−0.151 + 0.263i)17-s + 4.50i·18-s + (−0.702 − 0.405i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.980 + 0.195i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.980 + 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.52338 - 0.446266i\)
\(L(\frac12)\) \(\approx\) \(4.52338 - 0.446266i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.74 + 2.34i)T \)
good2 \( 1 + (-2.30 + 1.33i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.59 - 2.76i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 0.329iT - 5T^{2} \)
11 \( 1 + (-0.411 + 0.237i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.626 - 1.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.06 + 1.76i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.661 - 1.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.96 + 5.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.85iT - 31T^{2} \)
37 \( 1 + (4.04 - 2.33i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.89 + 2.82i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.42 - 4.20i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.30iT - 47T^{2} \)
53 \( 1 + 12.7T + 53T^{2} \)
59 \( 1 + (-1.48 - 0.857i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.91 + 5.04i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.47 + 4.89i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-10.6 - 6.17i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 9.20iT - 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 - 8.18iT - 83T^{2} \)
89 \( 1 + (9.43 - 5.44i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.0 - 7.55i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.76928592917504207833196707011, −9.961710735175865656061255336262, −9.296419084257662977538913663895, −8.100327748235625157776557825526, −6.59494659540519509397988790342, −5.30169379996254895950261679693, −4.77305733363060701860531959833, −3.85218548298780143302762970714, −3.09445029249555594299174218801, −2.25470570531290208929281247681, 2.00213357016344665510452575418, 2.93697650105619365373656671892, 4.04516951405321751045018434665, 5.23465781541107394732911421600, 6.38783425260673806699062040719, 6.89448970463631957975101569192, 7.59371711167947452777244116462, 8.366270745775411538961174650132, 9.239439824535658718520202497000, 11.18181798606537635581386091910

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