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Properties

Label 2-637-91.74-c1-0-13
Degree $2$
Conductor $637$
Sign $0.927 + 0.374i$
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  − 2.41·2-s + (−0.707 − 1.22i)3-s + 3.82·4-s + (1.91 + 3.31i)5-s + (1.70 + 2.95i)6-s − 4.41·8-s + (0.500 − 0.866i)9-s + (−4.62 − 8.00i)10-s + (−1.70 − 2.95i)11-s + (−2.70 − 4.68i)12-s + (3.5 + 0.866i)13-s + (2.70 − 4.68i)15-s + 2.99·16-s + 0.171·17-s + (−1.20 + 2.09i)18-s + (3 − 5.19i)19-s + ⋯
L(s)  = 1  − 1.70·2-s + (−0.408 − 0.707i)3-s + 1.91·4-s + (0.856 + 1.48i)5-s + (0.696 + 1.20i)6-s − 1.56·8-s + (0.166 − 0.288i)9-s + (−1.46 − 2.53i)10-s + (−0.514 − 0.891i)11-s + (−0.781 − 1.35i)12-s + (0.970 + 0.240i)13-s + (0.698 − 1.21i)15-s + 0.749·16-s + 0.0416·17-s + (−0.284 + 0.492i)18-s + (0.688 − 1.19i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.927 + 0.374i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.927 + 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.684763 - 0.133123i\)
\(L(\frac12)\) \(\approx\) \(0.684763 - 0.133123i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.5 - 0.866i)T \)
good2 \( 1 + 2.41T + 2T^{2} \)
3 \( 1 + (0.707 + 1.22i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.91 - 3.31i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.70 + 2.95i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 0.171T + 17T^{2} \)
19 \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.41T + 23T^{2} \)
29 \( 1 + (4.91 - 8.51i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.70 + 4.68i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.48T + 37T^{2} \)
41 \( 1 + (-2.91 + 5.04i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.292 + 0.507i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.82 - 6.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.75T + 59T^{2} \)
61 \( 1 + (-4.91 + 8.51i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.12 - 3.67i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.171 - 0.297i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.328 + 0.568i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.12 + 8.87i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.0T + 83T^{2} \)
89 \( 1 - 7.31T + 89T^{2} \)
97 \( 1 + (2.58 + 4.47i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.59220654955852996604930062439, −9.560774571263050315479759298989, −9.003010655589507963672413684130, −7.77251768708008196263479562709, −7.07346795670219371233675574604, −6.44438730854059947446506890014, −5.73264628236579170618527344802, −3.27367253939449758510185649677, −2.21455674358685738671728136838, −0.909420791150851291428313096137, 1.07613077015127637854158133368, 2.11896338093253648734131547997, 4.25306176559966417966256741878, 5.31508618830392141361147148249, 6.09538827223564718170453980482, 7.58572877656281436645469969708, 8.212318756230439205924996867118, 9.097684537225003801291807367840, 9.891783055517733343208804394526, 10.07374836439566415674937699725

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