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Properties

Label 2-637-7.4-c1-0-17
Degree $2$
Conductor $637$
Sign $-0.605 - 0.795i$
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  + (1.32 + 2.29i)2-s + (1.19 − 2.07i)3-s + (−2.52 + 4.37i)4-s + (1.82 + 3.16i)5-s + 6.36·6-s − 8.10·8-s + (−1.37 − 2.37i)9-s + (−4.85 + 8.40i)10-s + (−0.327 + 0.567i)11-s + (6.05 + 10.4i)12-s + 13-s + 8.75·15-s + (−5.70 − 9.88i)16-s + (1.19 − 2.07i)17-s + (3.63 − 6.30i)18-s + (−1.35 − 2.34i)19-s + ⋯
L(s)  = 1  + (0.938 + 1.62i)2-s + (0.691 − 1.19i)3-s + (−1.26 + 2.18i)4-s + (0.817 + 1.41i)5-s + 2.59·6-s − 2.86·8-s + (−0.456 − 0.791i)9-s + (−1.53 + 2.65i)10-s + (−0.0988 + 0.171i)11-s + (1.74 + 3.02i)12-s + 0.277·13-s + 2.26·15-s + (−1.42 − 2.47i)16-s + (0.290 − 0.503i)17-s + (0.857 − 1.48i)18-s + (−0.310 − 0.537i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33909 + 2.70126i\)
\(L(\frac12)\) \(\approx\) \(1.33909 + 2.70126i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good2 \( 1 + (-1.32 - 2.29i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.19 + 2.07i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.82 - 3.16i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.327 - 0.567i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.19 + 2.07i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.35 + 2.34i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.68 + 6.37i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.208T + 29T^{2} \)
31 \( 1 + (0.568 - 0.984i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.72 - 6.44i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 + 3.10T + 43T^{2} \)
47 \( 1 + (-2.30 - 3.98i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.62 - 4.55i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.12 + 7.15i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.948 + 1.64i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.44 + 11.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.75T + 71T^{2} \)
73 \( 1 + (-6.26 + 10.8i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.759 - 1.31i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.7T + 83T^{2} \)
89 \( 1 + (7.40 + 12.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.0T + 97T^{2} \)
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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

−10.98874369331260872979812931200, −9.731008586006313002753001951981, −8.628196276401745219391195245179, −7.79855568095427180711751615575, −7.15496490527734209652541508051, −6.45534730108446738766102693742, −6.01325808410659485710130239389, −4.62470290957021935952555385931, −3.18823143946371564697327092321, −2.39155427785449701940472030599, 1.31978081998146039087572749674, 2.47256243152806849617862955026, 3.83257658587863719919889198070, 4.20936416378128946540106961963, 5.37215047726054079495927231925, 5.78685725060117792074664621810, 8.271251076790563004188167919303, 9.092132218921289644163155241816, 9.669110103213298479157885117712, 10.16903479765052870592559821951

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