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Properties

Label 2-637-91.24-c1-0-2
Degree $2$
Conductor $637$
Sign $-0.931 - 0.362i$
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  + (1.33 − 0.357i)2-s + 1.07i·3-s + (−0.0814 + 0.0470i)4-s + (−2.77 − 0.742i)5-s + (0.383 + 1.42i)6-s + (−2.04 + 2.04i)8-s + 1.85·9-s − 3.96·10-s + (−1.00 + 1.00i)11-s + (−0.0504 − 0.0873i)12-s + (−3.54 + 0.634i)13-s + (0.796 − 2.97i)15-s + (−1.90 + 3.29i)16-s + (−2.09 − 3.63i)17-s + (2.46 − 0.661i)18-s + (−4.35 + 4.35i)19-s + ⋯
L(s)  = 1  + (0.942 − 0.252i)2-s + 0.618i·3-s + (−0.0407 + 0.0235i)4-s + (−1.23 − 0.332i)5-s + (0.156 + 0.583i)6-s + (−0.722 + 0.722i)8-s + 0.616·9-s − 1.25·10-s + (−0.304 + 0.304i)11-s + (−0.0145 − 0.0252i)12-s + (−0.984 + 0.176i)13-s + (0.205 − 0.767i)15-s + (−0.475 + 0.823i)16-s + (−0.509 − 0.881i)17-s + (0.581 − 0.155i)18-s + (−1.00 + 1.00i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 - 0.362i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.931 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.931 - 0.362i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (570, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.931 - 0.362i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.107780 + 0.573672i\)
\(L(\frac12)\) \(\approx\) \(0.107780 + 0.573672i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.54 - 0.634i)T \)
good2 \( 1 + (-1.33 + 0.357i)T + (1.73 - i)T^{2} \)
3 \( 1 - 1.07iT - 3T^{2} \)
5 \( 1 + (2.77 + 0.742i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.00 - 1.00i)T - 11iT^{2} \)
17 \( 1 + (2.09 + 3.63i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.35 - 4.35i)T - 19iT^{2} \)
23 \( 1 + (6.77 + 3.91i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.441 - 0.764i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.237 - 0.886i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.92 - 7.19i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-11.4 - 3.07i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.809 - 0.467i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.808 - 3.01i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.26 - 2.18i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.51 + 5.65i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + 0.0854iT - 61T^{2} \)
67 \( 1 + (-0.728 - 0.728i)T + 67iT^{2} \)
71 \( 1 + (-2.79 + 0.750i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.75 + 0.737i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.71 + 8.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.54 - 1.54i)T - 83iT^{2} \)
89 \( 1 + (-4.75 + 1.27i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.37 - 8.87i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.16186037244970552662690766369, −10.17218398676105711134441713316, −9.315862034962732406731308494671, −8.251433400001642582328192241964, −7.58914384736471452763004998068, −6.30428349317288004970701040628, −4.87930776645279836265306582783, −4.44272166258734124867143391748, −3.79783641433667482102130088241, −2.46444794248407444504487211814, 0.22391217178227392796507768655, 2.42598616908428980170224694764, 3.92274572520511644770073271965, 4.33679946175076154275771721363, 5.64422775554945616127332901759, 6.61542472456368551457509751344, 7.40402497101313512294761596905, 8.108261601763087404149187578610, 9.289409441648960233040559492892, 10.34191173365201560052596956508

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