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Properties

Label 2-637-91.74-c1-0-2
Degree $2$
Conductor $637$
Sign $-0.938 - 0.346i$
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  − 0.710·2-s + (1.20 + 2.08i)3-s − 1.49·4-s + (−0.644 − 1.11i)5-s + (−0.855 − 1.48i)6-s + 2.48·8-s + (−1.39 + 2.42i)9-s + (0.458 + 0.793i)10-s + (1.20 + 2.08i)11-s + (−1.80 − 3.11i)12-s + (−1.25 + 3.38i)13-s + (1.55 − 2.68i)15-s + 1.22·16-s − 3.90·17-s + (0.993 − 1.72i)18-s + (−2.94 + 5.10i)19-s + ⋯
L(s)  = 1  − 0.502·2-s + (0.695 + 1.20i)3-s − 0.747·4-s + (−0.288 − 0.499i)5-s + (−0.349 − 0.604i)6-s + 0.877·8-s + (−0.466 + 0.807i)9-s + (0.144 + 0.250i)10-s + (0.363 + 0.628i)11-s + (−0.519 − 0.900i)12-s + (−0.347 + 0.937i)13-s + (0.400 − 0.694i)15-s + 0.306·16-s − 0.946·17-s + (0.234 − 0.405i)18-s + (−0.675 + 1.17i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 - 0.346i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.938 - 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.938 - 0.346i$
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: \((2,\ 637,\ (\ :1/2),\ -0.938 - 0.346i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.122458 + 0.685879i\)
\(L(\frac12)\) \(\approx\) \(0.122458 + 0.685879i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.25 - 3.38i)T \)
good2 \( 1 + 0.710T + 2T^{2} \)
3 \( 1 + (-1.20 - 2.08i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.644 + 1.11i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.20 - 2.08i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 3.90T + 17T^{2} \)
19 \( 1 + (2.94 - 5.10i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.32T + 23T^{2} \)
29 \( 1 + (2.80 - 4.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.10 + 1.91i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.11T + 37T^{2} \)
41 \( 1 + (3.89 - 6.74i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.144 + 0.250i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.638 - 1.10i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.81 + 11.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 4.03T + 59T^{2} \)
61 \( 1 + (2.30 - 3.98i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.78 + 6.56i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.61 + 6.25i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.50 - 13.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.65 - 8.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.36T + 83T^{2} \)
89 \( 1 - 0.899T + 89T^{2} \)
97 \( 1 + (-7.83 - 13.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.42292356027733154879130306453, −9.987753656872776869266004246591, −9.153154119849167606797348725052, −8.668266362142760434262870561612, −7.923755547284931647347609406856, −6.62196433426724848133498322495, −5.04395934972325896332463669802, −4.27855069671710801421752945902, −3.81230374753687299254559905095, −1.93086060842979785920705861402, 0.41413489159715883492301660559, 2.01782401273565995792197596878, 3.23073195358911960678427612255, 4.46244013666255363444649085871, 5.87412334459095080227170248875, 7.03439337338526388654446535669, 7.58665545798443273038421997755, 8.568567829952437676493610328099, 8.885229538657902557069281035922, 10.15602849681741558532408491492

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