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Properties

Label 2-637-7.4-c1-0-28
Degree $2$
Conductor $637$
Sign $-0.386 + 0.922i$
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  + (−0.5 − 0.866i)2-s + (0.500 − 0.866i)4-s − 3·8-s + (1.5 + 2.59i)9-s + (1.5 − 2.59i)11-s + 13-s + (0.500 + 0.866i)16-s + (3.5 − 6.06i)17-s + (1.5 − 2.59i)18-s + (−3.5 − 6.06i)19-s − 3·22-s + (3 + 5.19i)23-s + (2.5 − 4.33i)25-s + (−0.5 − 0.866i)26-s − 5·29-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.250 − 0.433i)4-s − 1.06·8-s + (0.5 + 0.866i)9-s + (0.452 − 0.783i)11-s + 0.277·13-s + (0.125 + 0.216i)16-s + (0.848 − 1.47i)17-s + (0.353 − 0.612i)18-s + (−0.802 − 1.39i)19-s − 0.639·22-s + (0.625 + 1.08i)23-s + (0.5 − 0.866i)25-s + (−0.0980 − 0.169i)26-s − 0.928·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.386 + 0.922i$
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: \((2,\ 637,\ (\ :1/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.711665 - 1.06988i\)
\(L(\frac12)\) \(\approx\) \(0.711665 - 1.06988i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.5 + 6.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (-3.5 - 6.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.5 - 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5T + 71T^{2} \)
73 \( 1 + (-7 + 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−10.52744305472162628133655378335, −9.383676610850082375905670500794, −8.978064617523603637430001594600, −7.65220415425088187959310737100, −6.80102784244485052352179526989, −5.70699944540611899821201334029, −4.81098966747532589402369096897, −3.30887562154071612851098618355, −2.23190411749249809139188603495, −0.831276927847173280375774686574, 1.64588763118373417066028485251, 3.37986883239747882751581002366, 4.14720759242437579204427323377, 5.77211743066663245821371098304, 6.54522585179525654570302498294, 7.25806605948748998293713394908, 8.246256936157764178432145919971, 8.905146449742066753527586387736, 9.882074596398518959537477103817, 10.68458612053644865753977119388

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