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Properties

Label 2-637-91.54-c1-0-26
Degree $2$
Conductor $637$
Sign $0.160 + 0.987i$
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  + (−0.876 + 0.876i)2-s + (−1.92 − 1.11i)3-s + 0.463i·4-s + (2.51 − 0.674i)5-s + (2.66 − 0.713i)6-s + (−2.15 − 2.15i)8-s + (0.975 + 1.69i)9-s + (−1.61 + 2.79i)10-s + (1.36 − 0.365i)11-s + (0.515 − 0.893i)12-s + (−0.445 − 3.57i)13-s + (−5.60 − 1.50i)15-s + 2.85·16-s − 2.82·17-s + (−2.33 − 0.626i)18-s + (−1.61 + 6.04i)19-s + ⋯
L(s)  = 1  + (−0.619 + 0.619i)2-s + (−1.11 − 0.642i)3-s + 0.231i·4-s + (1.12 − 0.301i)5-s + (1.08 − 0.291i)6-s + (−0.763 − 0.763i)8-s + (0.325 + 0.563i)9-s + (−0.510 + 0.884i)10-s + (0.411 − 0.110i)11-s + (0.148 − 0.257i)12-s + (−0.123 − 0.992i)13-s + (−1.44 − 0.387i)15-s + 0.714·16-s − 0.685·17-s + (−0.550 − 0.147i)18-s + (−0.371 + 1.38i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.160 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.160 + 0.987i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.160 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.422614 - 0.359378i\)
\(L(\frac12)\) \(\approx\) \(0.422614 - 0.359378i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.445 + 3.57i)T \)
good2 \( 1 + (0.876 - 0.876i)T - 2iT^{2} \)
3 \( 1 + (1.92 + 1.11i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.51 + 0.674i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.36 + 0.365i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 + (1.61 - 6.04i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 1.01iT - 23T^{2} \)
29 \( 1 + (2.66 + 4.61i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.73 + 6.46i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (6.50 + 6.50i)T + 37iT^{2} \)
41 \( 1 + (-2.51 + 9.40i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.850 - 0.490i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.594 + 2.21i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.52 + 4.38i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.39 + 5.39i)T - 59iT^{2} \)
61 \( 1 + (-6.75 + 3.89i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.38 + 12.6i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.97 - 11.0i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (8.43 + 2.26i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (2.78 - 4.82i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.445 - 0.445i)T + 83iT^{2} \)
89 \( 1 + (0.108 - 0.108i)T - 89iT^{2} \)
97 \( 1 + (-2.87 + 0.771i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.23377227005196650503892709683, −9.472645628124974566102377630761, −8.556889196032203315473964732555, −7.65275291845966834994596586334, −6.71582027329031096740734447696, −5.96398979366731097536783122801, −5.49619548515756042002283029696, −3.81625317692945496243961227179, −2.02997927200871971899396606909, −0.42794084375390620616076714501, 1.49798748475690753036292985096, 2.67561913022065318766577842560, 4.52889979550355051888437500504, 5.28979473416648627164429404484, 6.26706000323094112457097130232, 6.83105349197503719056327925884, 8.760816236352604049136785388456, 9.298371168666533445476769819501, 10.11399132737692504723530310479, 10.63512251767908671002618335713

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