Properties

Label 2-637-91.59-c1-0-9
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

Related objects

Downloads

Learn more

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: $\chi_{637} (423, \cdot )$
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 \( 1 + (-2.87 - 0.771i)T + (84.0 + 48.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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