Properties

Label 2-637-91.16-c1-0-13
Degree $2$
Conductor $637$
Sign $0.230 - 0.972i$
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-s + (−0.707 + 1.22i)3-s − 4-s + (1.34 − 2.32i)5-s + (−0.707 + 1.22i)6-s − 3·8-s + (0.500 + 0.866i)9-s + (1.34 − 2.32i)10-s + (−2.89 + 5.01i)11-s + (0.707 − 1.22i)12-s + (2.75 + 2.32i)13-s + (1.89 + 3.28i)15-s − 16-s + 5.51·17-s + (0.500 + 0.866i)18-s + (1.41 + 2.44i)19-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.408 + 0.707i)3-s − 0.5·4-s + (0.600 − 1.03i)5-s + (−0.288 + 0.499i)6-s − 1.06·8-s + (0.166 + 0.288i)9-s + (0.424 − 0.735i)10-s + (−0.873 + 1.51i)11-s + (0.204 − 0.353i)12-s + (0.764 + 0.644i)13-s + (0.490 + 0.848i)15-s − 0.250·16-s + 1.33·17-s + (0.117 + 0.204i)18-s + (0.324 + 0.561i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.230 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.230 - 0.972i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.230 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23725 + 0.977971i\)
\(L(\frac12)\) \(\approx\) \(1.23725 + 0.977971i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.75 - 2.32i)T \)
good2 \( 1 - T + 2T^{2} \)
3 \( 1 + (0.707 - 1.22i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.34 + 2.32i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.89 - 5.01i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 5.51T + 17T^{2} \)
19 \( 1 + (-1.41 - 2.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.79T + 23T^{2} \)
29 \( 1 + (-4.39 - 7.61i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.707 - 1.22i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.79T + 37T^{2} \)
41 \( 1 + (4.87 + 8.44i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.897 + 1.55i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.41 - 2.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.29 + 5.70i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 1.12T + 59T^{2} \)
61 \( 1 + (0.779 + 1.34i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.89 - 5.01i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.90 - 5.02i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.89 + 10.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.89T + 83T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + (-2.12 + 3.67i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.49239166599129914673465808447, −9.967692652074767152227603553374, −9.192876948530416544520137210313, −8.351247033531228125959615865214, −7.11589709759761028009772883488, −5.64827132023306636168237167942, −5.12235509072512179965199004612, −4.59609480584769795615810620943, −3.47666847062815129371261462067, −1.63437641373743660032483214025, 0.789510091918761586153806048591, 2.89706855548721225433193763047, 3.47495647249182616856399331900, 5.13393294299939484461198862017, 6.02519488168574611549779174451, 6.33076216481192078467837663356, 7.67406441971494128090932492963, 8.509817418974811368871136409680, 9.714945560175529536453200131274, 10.47813705115508322413368105247

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