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Properties

Label 2-637-91.16-c1-0-41
Degree $2$
Conductor $637$
Sign $0.517 + 0.855i$
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  + 2.30·2-s + (1.08 − 1.87i)3-s + 3.30·4-s + (1.08 − 1.87i)5-s + (2.49 − 4.32i)6-s + 3.00·8-s + (−0.848 − 1.46i)9-s + (2.49 − 4.32i)10-s + (−2.45 + 4.25i)11-s + (3.57 − 6.19i)12-s + (1.41 + 3.31i)13-s + (−2.34 − 4.06i)15-s + 0.302·16-s − 7.15·17-s + (−1.95 − 3.38i)18-s + (1.08 + 1.87i)19-s + ⋯
L(s)  = 1  + 1.62·2-s + (0.625 − 1.08i)3-s + 1.65·4-s + (0.484 − 0.839i)5-s + (1.01 − 1.76i)6-s + 1.06·8-s + (−0.282 − 0.489i)9-s + (0.789 − 1.36i)10-s + (−0.739 + 1.28i)11-s + (1.03 − 1.78i)12-s + (0.391 + 0.920i)13-s + (−0.606 − 1.05i)15-s + 0.0756·16-s − 1.73·17-s + (−0.460 − 0.797i)18-s + (0.248 + 0.430i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.517 + 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.517 + 0.855i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.517 + 0.855i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.82001 - 2.15439i\)
\(L(\frac12)\) \(\approx\) \(3.82001 - 2.15439i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 - 3.31i)T \)
good2 \( 1 - 2.30T + 2T^{2} \)
3 \( 1 + (-1.08 + 1.87i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.08 + 1.87i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.45 - 4.25i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 7.15T + 17T^{2} \)
19 \( 1 + (-1.08 - 1.87i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.605T + 23T^{2} \)
29 \( 1 + (-1.15 - 1.99i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.57 + 6.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.60T + 37T^{2} \)
41 \( 1 + (4.99 + 8.64i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.25 + 10.8i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.755 - 1.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.19 + 2.07i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 2.82T + 59T^{2} \)
61 \( 1 + (-2.16 - 3.75i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2 - 3.46i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.16 + 3.75i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.30 + 5.72i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.82T + 83T^{2} \)
89 \( 1 + 6.50T + 89T^{2} \)
97 \( 1 + (6.83 - 11.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.75473291185574202753637050055, −9.386933651374478029644813280958, −8.623695382880687470022260417974, −7.37894274828187954901925717413, −6.82314430202803930276524857900, −5.77701547609468016565582207445, −4.80106129777003943672963717987, −4.03969752380130787483781398091, −2.37293705125911309265108419578, −1.89143819262262477746876585991, 2.76141597478575281667236073436, 3.00539110293087807965607387180, 4.14096778828151628742803287955, 5.01768060149473042878637678907, 6.01952698184159515872301130218, 6.67405848220992671874603117141, 8.121807115043959697203758334674, 9.056610307734375115193419430546, 10.13206185344224617196924687772, 11.00152284190360587136339955208

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