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Properties

Label 2-637-7.4-c1-0-2
Degree $2$
Conductor $637$
Sign $0.947 - 0.318i$
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  + (−0.916 − 1.58i)2-s + (−0.426 + 0.739i)3-s + (−0.678 + 1.17i)4-s + (−1.31 − 2.27i)5-s + 1.56·6-s − 1.17·8-s + (1.13 + 1.96i)9-s + (−2.40 + 4.16i)10-s + (−1.63 + 2.82i)11-s + (−0.579 − 1.00i)12-s + 13-s + 2.24·15-s + (2.43 + 4.21i)16-s + (−2.26 + 3.92i)17-s + (2.08 − 3.60i)18-s + (−2.03 − 3.52i)19-s + ⋯
L(s)  = 1  + (−0.647 − 1.12i)2-s + (−0.246 + 0.426i)3-s + (−0.339 + 0.587i)4-s + (−0.587 − 1.01i)5-s + 0.638·6-s − 0.416·8-s + (0.378 + 0.655i)9-s + (−0.760 + 1.31i)10-s + (−0.492 + 0.852i)11-s + (−0.167 − 0.289i)12-s + 0.277·13-s + 0.578·15-s + (0.609 + 1.05i)16-s + (−0.549 + 0.951i)17-s + (0.490 − 0.849i)18-s + (−0.466 − 0.807i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.947 - 0.318i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.947 - 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.534625 + 0.0874357i\)
\(L(\frac12)\) \(\approx\) \(0.534625 + 0.0874357i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good2 \( 1 + (0.916 + 1.58i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.426 - 0.739i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.31 + 2.27i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.63 - 2.82i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.26 - 3.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.03 + 3.52i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.26 - 3.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.42T + 29T^{2} \)
31 \( 1 + (-1.40 + 2.42i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.02 - 8.70i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.84T + 41T^{2} \)
43 \( 1 - 9.72T + 43T^{2} \)
47 \( 1 + (-4.72 - 8.18i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.63 - 4.55i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.28 + 2.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.59 - 9.68i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.990 + 1.71i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.7T + 71T^{2} \)
73 \( 1 + (6.06 - 10.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.95 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 + (5.33 + 9.23i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 13.7T + 97T^{2} \)
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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

−10.66727375602032309532119572126, −9.887773274778047307327009264173, −9.088132507738146869385558463188, −8.337943049046740270103748443983, −7.42549003792334634009896682539, −5.94472418801675103742996817644, −4.72342711191300913025281344522, −4.11445254624473257782654696059, −2.54336117366787452554649954586, −1.30152600034902563566890766782, 0.40944254555089205145594104064, 2.76446926278301755068372521098, 3.87656555381290900262113561066, 5.55234948566155981033516495921, 6.42087161343768356491003288259, 7.01425581312676131869436424815, 7.68333195388730364128032578342, 8.570240264153357836399391232448, 9.394059761534667019818444722772, 10.54784201179424010805006521272

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