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Properties

Label 2-637-91.34-c1-0-37
Degree $2$
Conductor $637$
Sign $-0.947 + 0.320i$
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  + (−0.508 − 0.508i)2-s − 1.66i·3-s − 1.48i·4-s + (1.36 − 1.36i)5-s + (−0.846 + 0.846i)6-s + (−1.77 + 1.77i)8-s + 0.230·9-s − 1.39·10-s + (2.25 − 2.25i)11-s − 2.46·12-s + (0.846 − 3.50i)13-s + (−2.27 − 2.27i)15-s − 1.16·16-s + 0.508·17-s + (−0.116 − 0.116i)18-s + (1.94 − 1.94i)19-s + ⋯
L(s)  = 1  + (−0.359 − 0.359i)2-s − 0.960i·3-s − 0.741i·4-s + (0.612 − 0.612i)5-s + (−0.345 + 0.345i)6-s + (−0.626 + 0.626i)8-s + 0.0766·9-s − 0.440·10-s + (0.680 − 0.680i)11-s − 0.712·12-s + (0.234 − 0.972i)13-s + (−0.588 − 0.588i)15-s − 0.291·16-s + 0.123·17-s + (−0.0275 − 0.0275i)18-s + (0.445 − 0.445i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.320i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.947 + 0.320i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (489, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.947 + 0.320i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.225644 - 1.37122i\)
\(L(\frac12)\) \(\approx\) \(0.225644 - 1.37122i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.846 + 3.50i)T \)
good2 \( 1 + (0.508 + 0.508i)T + 2iT^{2} \)
3 \( 1 + 1.66iT - 3T^{2} \)
5 \( 1 + (-1.36 + 1.36i)T - 5iT^{2} \)
11 \( 1 + (-2.25 + 2.25i)T - 11iT^{2} \)
17 \( 1 - 0.508T + 17T^{2} \)
19 \( 1 + (-1.94 + 1.94i)T - 19iT^{2} \)
23 \( 1 - 2.88iT - 23T^{2} \)
29 \( 1 + 2.40T + 29T^{2} \)
31 \( 1 + (-2.26 + 2.26i)T - 31iT^{2} \)
37 \( 1 + (6.88 - 6.88i)T - 37iT^{2} \)
41 \( 1 + (5.34 - 5.34i)T - 41iT^{2} \)
43 \( 1 - 12.5iT - 43T^{2} \)
47 \( 1 + (-7.89 - 7.89i)T + 47iT^{2} \)
53 \( 1 - 6.84T + 53T^{2} \)
59 \( 1 + (2.74 + 2.74i)T + 59iT^{2} \)
61 \( 1 + 6.37iT - 61T^{2} \)
67 \( 1 + (4.80 + 4.80i)T + 67iT^{2} \)
71 \( 1 + (1.90 + 1.90i)T + 71iT^{2} \)
73 \( 1 + (0.184 + 0.184i)T + 73iT^{2} \)
79 \( 1 - 5.56T + 79T^{2} \)
83 \( 1 + (-5.86 + 5.86i)T - 83iT^{2} \)
89 \( 1 + (8.66 + 8.66i)T + 89iT^{2} \)
97 \( 1 + (7.04 - 7.04i)T - 97iT^{2} \)
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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

−10.05207639024528525561213559859, −9.412857650884614642196459744826, −8.567861816758695044852284386172, −7.63727799046949171228506427912, −6.43983419173257032130876344105, −5.82004246684876749638247662616, −4.85721698181132046871371905181, −3.08897026197509509277923676070, −1.63515688290839169454338529747, −0.941308643889783443766296367105, 2.12789342273015274772869532071, 3.62477581375090336866278277446, 4.22039195134543968759786151785, 5.57365096933520648200423757506, 6.87381748276372527621355055097, 7.14700870844164369073588641803, 8.679831926681921659366122398787, 9.159881123097485823276042111085, 10.07592963920684335492861384298, 10.59141218933428397688851904219

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