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Properties

Label 2-637-13.9-c1-0-40
Degree $2$
Conductor $637$
Sign $-0.759 - 0.650i$
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  + (1.37 − 2.37i)2-s + (0.682 − 1.18i)3-s + (−2.75 − 4.77i)4-s − 0.741·5-s + (−1.87 − 3.23i)6-s − 9.63·8-s + (0.568 + 0.984i)9-s + (−1.01 + 1.75i)10-s + (0.682 − 1.18i)11-s − 7.52·12-s + (−0.301 − 3.59i)13-s + (−0.505 + 0.875i)15-s + (−7.68 + 13.3i)16-s + (−2.07 − 3.59i)17-s + 3.11·18-s + (3.63 + 6.29i)19-s + ⋯
L(s)  = 1  + (0.969 − 1.67i)2-s + (0.393 − 0.682i)3-s + (−1.37 − 2.38i)4-s − 0.331·5-s + (−0.763 − 1.32i)6-s − 3.40·8-s + (0.189 + 0.328i)9-s + (−0.321 + 0.556i)10-s + (0.205 − 0.356i)11-s − 2.17·12-s + (−0.0837 − 0.996i)13-s + (−0.130 + 0.226i)15-s + (−1.92 + 3.32i)16-s + (−0.503 − 0.871i)17-s + 0.734·18-s + (0.833 + 1.44i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.759 - 0.650i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.759 - 0.650i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.762547 + 2.06341i\)
\(L(\frac12)\) \(\approx\) \(0.762547 + 2.06341i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.301 + 3.59i)T \)
good2 \( 1 + (-1.37 + 2.37i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.682 + 1.18i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 0.741T + 5T^{2} \)
11 \( 1 + (-0.682 + 1.18i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.07 + 3.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.63 - 6.29i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.16 + 2.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.203 + 0.353i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.77T + 31T^{2} \)
37 \( 1 + (-3.05 + 5.28i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.627 + 1.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.870 - 1.50i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.85T + 47T^{2} \)
53 \( 1 - 4.56T + 53T^{2} \)
59 \( 1 + (5.49 + 9.51i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.26 - 5.65i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.87 + 11.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.40 - 4.17i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.06T + 73T^{2} \)
79 \( 1 + 9.12T + 79T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 + (0.880 - 1.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.76 - 8.25i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.30012852571718936892091227885, −9.564085038088246792906194387720, −8.451034861049961007390931655432, −7.47717704675796195618679621731, −6.05039965642475439124421087831, −5.16833185694247988756350613148, −4.08815482183325445595213917723, −3.10437485766477228733689439962, −2.17771911834640616534167367398, −0.889094019595360923006213974801, 3.02291289770240665294220758650, 4.17931013198800626209480635414, 4.46901946365493230978310616776, 5.69786823614856150267853896648, 6.75841896719518595269295087366, 7.25373525966047396377413370381, 8.398865348481333261636089588071, 9.066737817806298107753482545252, 9.784468720406299665393487216235, 11.43031918933861157304250091936

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