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Properties

Label 2-637-91.23-c1-0-20
Degree $2$
Conductor $637$
Sign $0.100 - 0.994i$
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  + 0.180i·2-s + (−0.913 + 1.58i)3-s + 1.96·4-s + (2.32 + 1.34i)5-s + (−0.285 − 0.165i)6-s + 0.717i·8-s + (−0.167 − 0.289i)9-s + (−0.242 + 0.420i)10-s + (2.33 + 1.34i)11-s + (−1.79 + 3.11i)12-s + (−1.92 − 3.05i)13-s + (−4.24 + 2.45i)15-s + 3.80·16-s + 4.76·17-s + (0.0523 − 0.0302i)18-s + (−0.163 + 0.0942i)19-s + ⋯
L(s)  = 1  + 0.127i·2-s + (−0.527 + 0.913i)3-s + 0.983·4-s + (1.04 + 0.600i)5-s + (−0.116 − 0.0673i)6-s + 0.253i·8-s + (−0.0557 − 0.0965i)9-s + (−0.0768 + 0.133i)10-s + (0.703 + 0.406i)11-s + (−0.518 + 0.898i)12-s + (−0.532 − 0.846i)13-s + (−1.09 + 0.633i)15-s + 0.951·16-s + 1.15·17-s + (0.0123 − 0.00712i)18-s + (−0.0374 + 0.0216i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.100 - 0.994i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.100 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43663 + 1.29826i\)
\(L(\frac12)\) \(\approx\) \(1.43663 + 1.29826i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.92 + 3.05i)T \)
good2 \( 1 - 0.180iT - 2T^{2} \)
3 \( 1 + (0.913 - 1.58i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.32 - 1.34i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.33 - 1.34i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 4.76T + 17T^{2} \)
19 \( 1 + (0.163 - 0.0942i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.39T + 23T^{2} \)
29 \( 1 + (3.54 + 6.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.20 + 1.84i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.95iT - 37T^{2} \)
41 \( 1 + (4.70 - 2.71i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.00 - 6.93i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.60 - 0.924i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.53 - 6.12i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 7.58iT - 59T^{2} \)
61 \( 1 + (0.205 + 0.356i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.87 + 5.70i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.89 - 1.67i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (12.3 - 7.10i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.55 - 7.89i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16.5iT - 83T^{2} \)
89 \( 1 + 5.89iT - 89T^{2} \)
97 \( 1 + (0.390 + 0.225i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.50189096358056007339228461179, −10.09476879749026394474290943066, −9.558692247656137679205457744352, −7.942966526248657916488356131453, −7.16945030727794154261657096700, −5.89914850488136295922776927744, −5.75641323198267601627367873206, −4.32839789439607123005496722680, −2.99792996025679915174243461491, −1.85803418211040884993124184432, 1.28745949726848204392697719818, 1.95163065188968619316251892674, 3.52451281481232884032536646945, 5.21498287645318680530969054112, 6.04285816367580219458542957745, 6.69018602748845563738281486292, 7.44076890822882408213912503797, 8.643941249289327404587058607179, 9.681088369582578873680964007047, 10.32029872875202401772598148194

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