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Properties

Label 2-637-91.81-c1-0-17
Degree $2$
Conductor $637$
Sign $-0.703 - 0.710i$
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  + (−1.30 + 2.26i)2-s + 2.61·3-s + (−2.42 − 4.20i)4-s + (1.30 + 2.26i)5-s + (−3.42 + 5.93i)6-s + 7.47·8-s + 3.85·9-s − 6.85·10-s + 1.85·11-s + (−6.35 − 11.0i)12-s + (2.5 + 2.59i)13-s + (3.42 + 5.93i)15-s + (−4.92 + 8.53i)16-s + (−0.736 − 1.27i)17-s + (−5.04 + 8.73i)18-s + 1.85·19-s + ⋯
L(s)  = 1  + (−0.925 + 1.60i)2-s + 1.51·3-s + (−1.21 − 2.10i)4-s + (0.585 + 1.01i)5-s + (−1.39 + 2.42i)6-s + 2.64·8-s + 1.28·9-s − 2.16·10-s + 0.559·11-s + (−1.83 − 3.17i)12-s + (0.693 + 0.720i)13-s + (0.884 + 1.53i)15-s + (−1.23 + 2.13i)16-s + (−0.178 − 0.309i)17-s + (−1.18 + 2.05i)18-s + 0.425·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.703 - 0.710i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.703 - 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.636778 + 1.52753i\)
\(L(\frac12)\) \(\approx\) \(0.636778 + 1.52753i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 2.59i)T \)
good2 \( 1 + (1.30 - 2.26i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 - 2.61T + 3T^{2} \)
5 \( 1 + (-1.30 - 2.26i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 1.85T + 11T^{2} \)
17 \( 1 + (0.736 + 1.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 1.85T + 19T^{2} \)
23 \( 1 + (-2.23 + 3.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.54 + 6.14i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.35 - 4.07i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.381 - 0.661i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.28 - 10.8i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.11 + 1.93i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.88 - 3.25i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.11 + 1.93i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + 12.7T + 67T^{2} \)
71 \( 1 + (-7.09 + 12.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.70T + 83T^{2} \)
89 \( 1 + (-2.45 + 4.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.42 + 16.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.28996218858614246679134377446, −9.586001831193624235035254269305, −8.991032833651603273647600975065, −8.318346186972984222965618043116, −7.42160219920500983959718446407, −6.70509628044687772080075216176, −6.06284926775233439740003444822, −4.55865679378103979402191534661, −3.15798453148614947837328791181, −1.72636254295764892633252070540, 1.23235897296191225582736320914, 2.04132488132278653363383979846, 3.29760388512439541783887985672, 3.89385194790681835912585836718, 5.36093236102240234558330182428, 7.33245524479565063382594329955, 8.252100945519466762027465839438, 8.918558459437281776913800228591, 9.216629185926511914001182310232, 10.00891085542311033615999304552

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