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Properties

Label 2-637-91.34-c1-0-5
Degree $2$
Conductor $637$
Sign $0.751 - 0.659i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.751 - 0.659i$
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: \((2,\ 637,\ (\ :1/2),\ 0.751 - 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.516361 + 0.194556i\)
\(L(\frac12)\) \(\approx\) \(0.516361 + 0.194556i\)
\(L(\frac{3}{2})\) 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

−10.55324445776515819873777057022, −9.907640025632004536106676279820, −9.142091721467346165344061342974, −8.280036387069956397954644617491, −7.24901255230294763220325635625, −6.40318366490764239096376907290, −5.06983419880539021527192458367, −4.00645239732542459543810373147, −2.31402415115968236560888028121, −1.61887219315446365236471154676, 0.38177091433939181527755961229, 2.61838026643141044877689214088, 3.92209664229124323239111789352, 5.10039466348803988517232976330, 6.29575347465675439375146581021, 6.96910751662713143118879600359, 8.046075251020989714248569205431, 8.547076346896244957278624005490, 9.432604475893900450088938432185, 10.44711739219864714204674206482

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