Properties

Label 2-637-13.10-c1-0-32
Degree $2$
Conductor $637$
Sign $-0.800 + 0.599i$
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  + (−1.16 + 0.672i)2-s + (−1.02 − 1.77i)3-s + (−0.0951 + 0.164i)4-s − 3.56i·5-s + (2.38 + 1.37i)6-s − 2.94i·8-s + (−0.601 + 1.04i)9-s + (2.39 + 4.15i)10-s + (1.10 − 0.639i)11-s + 0.390·12-s + (3.57 − 0.474i)13-s + (−6.33 + 3.65i)15-s + (1.79 + 3.10i)16-s + (3.86 − 6.70i)17-s − 1.61i·18-s + (−0.817 − 0.471i)19-s + ⋯
L(s)  = 1  + (−0.823 + 0.475i)2-s + (−0.591 − 1.02i)3-s + (−0.0475 + 0.0824i)4-s − 1.59i·5-s + (0.975 + 0.562i)6-s − 1.04i·8-s + (−0.200 + 0.347i)9-s + (0.758 + 1.31i)10-s + (0.333 − 0.192i)11-s + 0.112·12-s + (0.991 − 0.131i)13-s + (−1.63 + 0.944i)15-s + (0.447 + 0.775i)16-s + (0.938 − 1.62i)17-s − 0.381i·18-s + (−0.187 − 0.108i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.800 + 0.599i$
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: \((2,\ 637,\ (\ :1/2),\ -0.800 + 0.599i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.205567 - 0.617472i\)
\(L(\frac12)\) \(\approx\) \(0.205567 - 0.617472i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.57 + 0.474i)T \)
good2 \( 1 + (1.16 - 0.672i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (1.02 + 1.77i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 3.56iT - 5T^{2} \)
11 \( 1 + (-1.10 + 0.639i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.86 + 6.70i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.817 + 0.471i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.823 - 1.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.02 + 3.50i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.15iT - 31T^{2} \)
37 \( 1 + (0.914 - 0.528i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.63 - 2.09i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.91 + 3.31i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.894iT - 47T^{2} \)
53 \( 1 + 0.0799T + 53T^{2} \)
59 \( 1 + (9.68 + 5.59i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.81 + 6.60i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.47 - 3.16i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.89 - 5.71i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 0.760iT - 73T^{2} \)
79 \( 1 + 2.85T + 79T^{2} \)
83 \( 1 - 2.32iT - 83T^{2} \)
89 \( 1 + (6.56 - 3.78i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.414 + 0.239i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.825655279394923378567397429591, −9.153544060734687936537104789678, −8.433667609652726562536955892420, −7.68868768889830368871473482957, −6.86410948928328384156563295167, −5.88867631596088277382800507864, −4.93336892810609225898046287336, −3.61588769158557631444785158298, −1.33922418524071962487516063158, −0.58370566254541459154089918732, 1.76821791425778012131605262903, 3.29016319467417342174134163115, 4.23193196543279531591214640484, 5.66388964594785717627056253772, 6.25974481754495449095132018702, 7.54852073963263819732187140551, 8.553851129010096710350984606545, 9.599294115265492034910700630808, 10.24114363670030209220679128003, 10.83230813455755828398984450939

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