Properties

Label 2-637-91.4-c1-0-40
Degree $2$
Conductor $637$
Sign $-0.372 - 0.927i$
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  − 1.73i·2-s + (−0.5 − 0.866i)3-s − 0.999·4-s + (−1.5 + 0.866i)5-s + (−1.49 + 0.866i)6-s − 1.73i·8-s + (1 − 1.73i)9-s + (1.49 + 2.59i)10-s + (−4.5 + 2.59i)11-s + (0.499 + 0.866i)12-s + (1 − 3.46i)13-s + (1.5 + 0.866i)15-s − 5·16-s − 6·17-s + (−3 − 1.73i)18-s + (1.5 + 0.866i)19-s + ⋯
L(s)  = 1  − 1.22i·2-s + (−0.288 − 0.499i)3-s − 0.499·4-s + (−0.670 + 0.387i)5-s + (−0.612 + 0.353i)6-s − 0.612i·8-s + (0.333 − 0.577i)9-s + (0.474 + 0.821i)10-s + (−1.35 + 0.783i)11-s + (0.144 + 0.249i)12-s + (0.277 − 0.960i)13-s + (0.387 + 0.223i)15-s − 1.25·16-s − 1.45·17-s + (−0.707 − 0.408i)18-s + (0.344 + 0.198i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.372 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.372 - 0.927i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.372 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.267371 + 0.395611i\)
\(L(\frac12)\) \(\approx\) \(0.267371 + 0.395611i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 3.46i)T \)
good2 \( 1 + 1.73iT - 2T^{2} \)
3 \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.5 - 2.59i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + (-1.5 - 0.866i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.5 - 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (-4.5 - 2.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (7.5 - 4.33i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 3.46iT - 59T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.5 + 4.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.5 + 0.866i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.5 + 4.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 - 6.92iT - 89T^{2} \)
97 \( 1 + (-4.5 + 2.59i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.24681892052838759436974869280, −9.485150478624857185707624148994, −8.187769139952625408806874395660, −7.27285222135996009362554055102, −6.59635508802287782532496163457, −5.18780692382171607647907423357, −3.95933413418475588613507594495, −3.02418231469773764169373465398, −1.87852419102190085795320780970, −0.25148432323485417672786791210, 2.40279497122692555087689854232, 4.19914138268499238712459652604, 4.87230657305329020076787892974, 5.77607764717510825153228139047, 6.75416339491388564675262158490, 7.72100756015289648089441662630, 8.297475241199766100435902232246, 9.134862506305999947036118212250, 10.37087380128518449082953256803, 11.22783433584987940958054786872

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