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Properties

Label 2-637-13.12-c1-0-9
Degree $2$
Conductor $637$
Sign $0.159 + 0.987i$
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  − 2.73i·2-s − 1.15·3-s − 5.47·4-s + 1.87i·5-s + 3.16i·6-s + 9.50i·8-s − 1.66·9-s + 5.13·10-s − 2.29i·11-s + 6.32·12-s + (0.574 + 3.55i)13-s − 2.17i·15-s + 15.0·16-s + 6.07·17-s + 4.54i·18-s − 5.15i·19-s + ⋯
L(s)  = 1  − 1.93i·2-s − 0.667·3-s − 2.73·4-s + 0.839i·5-s + 1.29i·6-s + 3.36i·8-s − 0.554·9-s + 1.62·10-s − 0.693i·11-s + 1.82·12-s + (0.159 + 0.987i)13-s − 0.560i·15-s + 3.75·16-s + 1.47·17-s + 1.07i·18-s − 1.18i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.159 + 0.987i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (246, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.159 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.637963 - 0.543294i\)
\(L(\frac12)\) \(\approx\) \(0.637963 - 0.543294i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.574 - 3.55i)T \)
good2 \( 1 + 2.73iT - 2T^{2} \)
3 \( 1 + 1.15T + 3T^{2} \)
5 \( 1 - 1.87iT - 5T^{2} \)
11 \( 1 + 2.29iT - 11T^{2} \)
17 \( 1 - 6.07T + 17T^{2} \)
19 \( 1 + 5.15iT - 19T^{2} \)
23 \( 1 + 4.41T + 23T^{2} \)
29 \( 1 - 7.50T + 29T^{2} \)
31 \( 1 - 4.33iT - 31T^{2} \)
37 \( 1 - 3.16iT - 37T^{2} \)
41 \( 1 - 2.45iT - 41T^{2} \)
43 \( 1 + 2.17T + 43T^{2} \)
47 \( 1 - 8.90iT - 47T^{2} \)
53 \( 1 - 8.10T + 53T^{2} \)
59 \( 1 - 7.13iT - 59T^{2} \)
61 \( 1 - 8.00T + 61T^{2} \)
67 \( 1 + 1.15iT - 67T^{2} \)
71 \( 1 + 5.11iT - 71T^{2} \)
73 \( 1 - 1.96iT - 73T^{2} \)
79 \( 1 - 3.81T + 79T^{2} \)
83 \( 1 - 2.49iT - 83T^{2} \)
89 \( 1 - 10.4iT - 89T^{2} \)
97 \( 1 + 2.51iT - 97T^{2} \)
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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

−10.62898546016706751797468200541, −9.965659475867636585504962958163, −8.969879806267961687319503461256, −8.185502627387466853057757160883, −6.61793539134826405978394964294, −5.52732000047735720067099375231, −4.54024499440299996302027636674, −3.31572766332886116019163232882, −2.63552976307937293773089309113, −1.00361730339868902955540741875, 0.69807622465968078829108985231, 3.71577362885389210265891286890, 4.88734952082295181356137531481, 5.54808771991531748870101421809, 6.06540395063971917865312966778, 7.21533323753488757464986102070, 8.169233880583607707906417042066, 8.493046846287920259638967582829, 9.775833736533140278996837123933, 10.29066648869050193299171646099

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