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Properties

Label 2-637-91.74-c1-0-38
Degree $2$
Conductor $637$
Sign $-0.515 + 0.856i$
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-s + (0.707 + 1.22i)3-s − 4-s + (−1.34 − 2.32i)5-s + (0.707 + 1.22i)6-s − 3·8-s + (0.500 − 0.866i)9-s + (−1.34 − 2.32i)10-s + (−2.89 − 5.01i)11-s + (−0.707 − 1.22i)12-s + (−2.75 + 2.32i)13-s + (1.89 − 3.28i)15-s − 16-s − 5.51·17-s + (0.500 − 0.866i)18-s + (−1.41 + 2.44i)19-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.408 + 0.707i)3-s − 0.5·4-s + (−0.600 − 1.03i)5-s + (0.288 + 0.499i)6-s − 1.06·8-s + (0.166 − 0.288i)9-s + (−0.424 − 0.735i)10-s + (−0.873 − 1.51i)11-s + (−0.204 − 0.353i)12-s + (−0.764 + 0.644i)13-s + (0.490 − 0.848i)15-s − 0.250·16-s − 1.33·17-s + (0.117 − 0.204i)18-s + (−0.324 + 0.561i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.515 + 0.856i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.515 + 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.429791 - 0.760641i\)
\(L(\frac12)\) \(\approx\) \(0.429791 - 0.760641i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.75 - 2.32i)T \)
good2 \( 1 - T + 2T^{2} \)
3 \( 1 + (-0.707 - 1.22i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.34 + 2.32i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.89 + 5.01i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 5.51T + 17T^{2} \)
19 \( 1 + (1.41 - 2.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.79T + 23T^{2} \)
29 \( 1 + (-4.39 + 7.61i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.707 - 1.22i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.79T + 37T^{2} \)
41 \( 1 + (-4.87 + 8.44i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.897 - 1.55i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.41 - 2.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.29 - 5.70i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 1.12T + 59T^{2} \)
61 \( 1 + (-0.779 + 1.34i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.89 + 5.01i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.90 - 5.02i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.89 - 10.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
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

−10.21866551777558510528680245419, −9.103422062060868905927373225047, −8.792849163150022904416982323763, −7.975976368759703783235259502844, −6.44539905399717396934411686817, −5.35446069998852230239066953627, −4.48161847172869964329144989746, −3.98377759264676169751721074122, −2.81479634220447093534516221044, −0.34712572860200048137294165223, 2.31825306463554068587329085815, 3.07111303589758090911808126105, 4.51043433773061834841750889285, 5.07226913160353521112506525430, 6.70798215203037053912402091449, 7.19642893809106444235468755374, 8.025266706393601400359786592564, 9.034856273411009291801896572138, 10.18789617960278926103584504479, 10.82376252824302818917111853914

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