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Properties

Label 2-637-13.4-c1-0-11
Degree $2$
Conductor $637$
Sign $-0.964 - 0.265i$
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 + (−0.895 + 1.55i)3-s + (1.39 + 2.41i)4-s + 2.18i·5-s + (−3.39 + 1.96i)6-s + 1.73i·8-s + (−0.104 − 0.180i)9-s + (−2.39 + 4.14i)10-s + (−1.10 − 0.637i)11-s − 4.99·12-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 − 0.456i·18-s + (5.68 − 3.28i)19-s + ⋯
L(s)  = 1  + (1.34 + 0.773i)2-s + (−0.517 + 0.895i)3-s + (0.697 + 1.20i)4-s + 0.978i·5-s + (−1.38 + 0.800i)6-s + 0.612i·8-s + (−0.0347 − 0.0602i)9-s + (−0.757 + 1.31i)10-s + (−0.332 − 0.192i)11-s − 1.44·12-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 − 0.107i·18-s + (1.30 − 0.753i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.964 - 0.265i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.964 - 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.321450 + 2.38260i\)
\(L(\frac12)\) \(\approx\) \(0.321450 + 2.38260i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.895 - 1.55i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 2.18iT - 5T^{2} \)
11 \( 1 + (1.10 + 0.637i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.68 + 3.28i)T + (9.5 - 16.4i)T^{2} \)
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 \( 1 + 8.66iT - 31T^{2} \)
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 \( 1 - 4.28iT - 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 + (-7.66 + 4.42i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.37 - 11.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.87 - 5.70i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.791 - 0.456i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 3.46iT - 73T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
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

−11.32516336337575174425655259730, −9.967549535086978706515310635288, −9.729257188213040172826199697249, −7.79232937711489151074910999446, −7.28464483994551031939350551727, −6.16473037672556940125161873214, −5.47607828098225386784987627837, −4.64977096967864172658224527500, −3.74252111441710993804958462428, −2.72149990093153078315780972755, 0.946768860521896181542105793764, 2.21130596328021549322311126868, 3.48497073732612541650986951648, 4.82805757096900479260107752463, 5.24256810482218738934865840794, 6.27429343856061941558013860965, 7.34150167552812145586146284221, 8.289297613920826988663679709257, 9.595415179604265629080156151602, 10.43663376236652240744401618521

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