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Properties

Label 2-637-13.4-c1-0-25
Degree $2$
Conductor $637$
Sign $-0.204 - 0.978i$
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  + (2.24 + 1.29i)2-s + (0.259 − 0.449i)3-s + (2.35 + 4.07i)4-s + 1.61i·5-s + (1.16 − 0.671i)6-s + 6.99i·8-s + (1.36 + 2.36i)9-s + (−2.08 + 3.61i)10-s + (−2.34 − 1.35i)11-s + 2.43·12-s + (−2.36 − 2.71i)13-s + (0.723 + 0.417i)15-s + (−4.34 + 7.53i)16-s + (−1.56 − 2.70i)17-s + 7.06i·18-s + (3.18 − 1.84i)19-s + ⋯
L(s)  = 1  + (1.58 + 0.915i)2-s + (0.149 − 0.259i)3-s + (1.17 + 2.03i)4-s + 0.720i·5-s + (0.474 − 0.273i)6-s + 2.47i·8-s + (0.455 + 0.788i)9-s + (−0.659 + 1.14i)10-s + (−0.706 − 0.407i)11-s + 0.703·12-s + (−0.656 − 0.753i)13-s + (0.186 + 0.107i)15-s + (−1.08 + 1.88i)16-s + (−0.379 − 0.656i)17-s + 1.66i·18-s + (0.731 − 0.422i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.204 - 0.978i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.204 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.30565 + 2.83694i\)
\(L(\frac12)\) \(\approx\) \(2.30565 + 2.83694i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.36 + 2.71i)T \)
good2 \( 1 + (-2.24 - 1.29i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.259 + 0.449i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 1.61iT - 5T^{2} \)
11 \( 1 + (2.34 + 1.35i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.56 + 2.70i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.18 + 1.84i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.993 + 1.71i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.68 + 4.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 10.4iT - 31T^{2} \)
37 \( 1 + (5.15 + 2.97i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.66 - 3.85i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.67 + 2.90i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.05iT - 47T^{2} \)
53 \( 1 - 7.26T + 53T^{2} \)
59 \( 1 + (9.89 - 5.71i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.46 + 2.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.7 + 6.79i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.17 + 0.675i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 9.10iT - 73T^{2} \)
79 \( 1 + 6.20T + 79T^{2} \)
83 \( 1 + 2.69iT - 83T^{2} \)
89 \( 1 + (-1.52 - 0.879i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.4 + 7.74i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.94889936913920250498405688936, −10.29099833244912192693028677875, −8.673627605562486165461133767960, −7.54527340033563107594501048648, −7.28746605097276589224816321426, −6.30634509929335039595651249823, −5.20182891720919562860752035389, −4.68991950144631633737978925342, −3.15236968530946492173716177039, −2.59969487735756121153742345515, 1.40321168277685861340663715578, 2.66860435980687688985882495212, 3.85923712741336547902442397549, 4.56469204998303198083760239884, 5.34453435666525195133465253412, 6.37995118320729870138437638966, 7.43103134654419929563101627195, 8.937877342612808924097035570318, 9.781535898163142529864188949865, 10.47024529547908721894343098463

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