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Properties

Label 2-637-13.3-c1-0-27
Degree $2$
Conductor $637$
Sign $-0.566 + 0.824i$
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 + (−0.707 − 1.22i)3-s + (0.500 − 0.866i)4-s + 4.09·5-s + (−0.707 + 1.22i)6-s − 3·8-s + (0.500 − 0.866i)9-s + (−2.04 − 3.54i)10-s + (1.89 + 3.28i)11-s − 1.41·12-s + (−0.634 − 3.54i)13-s + (−2.89 − 5.01i)15-s + (0.500 + 0.866i)16-s + (0.634 − 1.09i)17-s − 1.00·18-s + (1.41 − 2.44i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.408 − 0.707i)3-s + (0.250 − 0.433i)4-s + 1.83·5-s + (−0.288 + 0.499i)6-s − 1.06·8-s + (0.166 − 0.288i)9-s + (−0.647 − 1.12i)10-s + (0.572 + 0.991i)11-s − 0.408·12-s + (−0.176 − 0.984i)13-s + (−0.748 − 1.29i)15-s + (0.125 + 0.216i)16-s + (0.153 − 0.266i)17-s − 0.235·18-s + (0.324 − 0.561i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.566 + 0.824i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.566 + 0.824i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.746473 - 1.41851i\)
\(L(\frac12)\) \(\approx\) \(0.746473 - 1.41851i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.634 + 3.54i)T \)
good2 \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.707 + 1.22i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 4.09T + 5T^{2} \)
11 \( 1 + (-1.89 - 3.28i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.634 + 1.09i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.41 + 2.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.89 - 6.75i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.397 + 0.689i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.41T + 31T^{2} \)
37 \( 1 + (1.39 + 2.42i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.48 + 2.57i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.89 - 6.75i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + 12.5T + 53T^{2} \)
59 \( 1 + (-6.21 + 10.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.17 - 7.22i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.89 - 3.28i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 12.5T + 73T^{2} \)
79 \( 1 + 2.20T + 79T^{2} \)
83 \( 1 + 9.89T + 83T^{2} \)
89 \( 1 + (-7.48 - 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.12 + 3.67i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.08662103297143632092617027769, −9.620886269399728999300259177659, −9.073796135688037548811002529183, −7.33264529646235685025673298359, −6.62565652719221107625113687229, −5.80302163376652866964315317000, −5.13515405868487536186666554242, −3.03905272948381252521085574597, −1.88022673012806132182412890956, −1.14423458857019977662047049032, 1.83045509365056935015039331927, 3.17073437718250360058143175614, 4.63446522518492835427187290604, 5.72289784736615186666225095090, 6.29832997321078312210044553898, 7.12671152473831952617863435905, 8.540418093346059376237589206280, 9.106600271988748835685429285810, 9.919787923764635484117936017566, 10.67071817006721559873442472223

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