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Properties

Label 2-637-91.83-c1-0-11
Degree $2$
Conductor $637$
Sign $0.545 - 0.837i$
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  + (−1.12 + 1.12i)2-s − 0.503i·3-s − 0.525i·4-s + (−0.0563 − 0.0563i)5-s + (0.565 + 0.565i)6-s + (−1.65 − 1.65i)8-s + 2.74·9-s + 0.126·10-s + (−2.98 − 2.98i)11-s − 0.264·12-s + (−0.565 + 3.56i)13-s + (−0.0283 + 0.0283i)15-s + 4.77·16-s + 5.80·17-s + (−3.08 + 3.08i)18-s + (3.74 + 3.74i)19-s + ⋯
L(s)  = 1  + (−0.794 + 0.794i)2-s − 0.290i·3-s − 0.262i·4-s + (−0.0251 − 0.0251i)5-s + (0.231 + 0.231i)6-s + (−0.585 − 0.585i)8-s + 0.915·9-s + 0.0400·10-s + (−0.901 − 0.901i)11-s − 0.0763·12-s + (−0.156 + 0.987i)13-s + (−0.00732 + 0.00732i)15-s + 1.19·16-s + 1.40·17-s + (−0.727 + 0.727i)18-s + (0.858 + 0.858i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.545 - 0.837i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (538, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.545 - 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.865882 + 0.469374i\)
\(L(\frac12)\) \(\approx\) \(0.865882 + 0.469374i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.565 - 3.56i)T \)
good2 \( 1 + (1.12 - 1.12i)T - 2iT^{2} \)
3 \( 1 + 0.503iT - 3T^{2} \)
5 \( 1 + (0.0563 + 0.0563i)T + 5iT^{2} \)
11 \( 1 + (2.98 + 2.98i)T + 11iT^{2} \)
17 \( 1 - 5.80T + 17T^{2} \)
19 \( 1 + (-3.74 - 3.74i)T + 19iT^{2} \)
23 \( 1 + 0.872iT - 23T^{2} \)
29 \( 1 - 0.362T + 29T^{2} \)
31 \( 1 + (-0.986 - 0.986i)T + 31iT^{2} \)
37 \( 1 + (-2.75 - 2.75i)T + 37iT^{2} \)
41 \( 1 + (-7.70 - 7.70i)T + 41iT^{2} \)
43 \( 1 - 2.65iT - 43T^{2} \)
47 \( 1 + (-2.04 + 2.04i)T - 47iT^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 + (1.56 - 1.56i)T - 59iT^{2} \)
61 \( 1 - 4.19iT - 61T^{2} \)
67 \( 1 + (7.15 - 7.15i)T - 67iT^{2} \)
71 \( 1 + (-3.65 + 3.65i)T - 71iT^{2} \)
73 \( 1 + (8.41 - 8.41i)T - 73iT^{2} \)
79 \( 1 - 8.55T + 79T^{2} \)
83 \( 1 + (4.91 + 4.91i)T + 83iT^{2} \)
89 \( 1 + (-5.69 + 5.69i)T - 89iT^{2} \)
97 \( 1 + (6.04 + 6.04i)T + 97iT^{2} \)
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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

−10.25890442296754805419404464506, −9.840683105686597171910182111493, −8.777543312044922069010874187480, −7.890900460769259008371218264678, −7.47648439400685460361518453980, −6.43339178124821656562092408937, −5.62204143521253140184500866126, −4.18889447165761700715912956271, −2.93712398426194290683091187445, −1.06391189054670928651654994962, 0.958998286017459804035652128803, 2.37806558781703693767305929412, 3.47570664688950206583774674684, 4.98573315454435478928110793304, 5.65356944057037863669528485922, 7.37360833753461363304146564736, 7.75667752065926384587931624885, 9.128579703163180944257968071839, 9.737663973641907436722516780338, 10.35715549198073521214245104511

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