Properties

Label 2-639-213.62-c1-0-5
Degree $2$
Conductor $639$
Sign $-0.159 - 0.987i$
Analytic cond. $5.10244$
Root an. cond. $2.25885$
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  + (−1.26 + 0.171i)2-s + (−0.357 + 0.0987i)4-s + (1.07 + 1.48i)5-s + (3.41 + 3.26i)7-s + (2.78 − 1.18i)8-s + (−1.61 − 1.69i)10-s + (−2.58 − 0.232i)11-s + (0.360 + 4.00i)13-s + (−4.88 − 3.54i)14-s + (−2.67 + 1.60i)16-s + (0.0523 + 0.161i)17-s + (6.75 − 2.53i)19-s + (−0.533 − 0.425i)20-s + (3.30 − 0.148i)22-s + (−5.21 − 2.51i)23-s + ⋯
L(s)  = 1  + (−0.894 + 0.121i)2-s + (−0.178 + 0.0493i)4-s + (0.482 + 0.664i)5-s + (1.29 + 1.23i)7-s + (0.983 − 0.420i)8-s + (−0.512 − 0.535i)10-s + (−0.778 − 0.0700i)11-s + (0.0998 + 1.10i)13-s + (−1.30 − 0.947i)14-s + (−0.669 + 0.400i)16-s + (0.0126 + 0.0390i)17-s + (1.54 − 0.581i)19-s + (−0.119 − 0.0950i)20-s + (0.704 − 0.0316i)22-s + (−1.08 − 0.524i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.159 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(639\)    =    \(3^{2} \cdot 71\)
Sign: $-0.159 - 0.987i$
Analytic conductor: \(5.10244\)
Root analytic conductor: \(2.25885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{639} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 639,\ (\ :1/2),\ -0.159 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.630983 + 0.741406i\)
\(L(\frac12)\) \(\approx\) \(0.630983 + 0.741406i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
71 \( 1 + (-5.69 - 6.21i)T \)
good2 \( 1 + (1.26 - 0.171i)T + (1.92 - 0.532i)T^{2} \)
5 \( 1 + (-1.07 - 1.48i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (-3.41 - 3.26i)T + (0.314 + 6.99i)T^{2} \)
11 \( 1 + (2.58 + 0.232i)T + (10.8 + 1.96i)T^{2} \)
13 \( 1 + (-0.360 - 4.00i)T + (-12.7 + 2.32i)T^{2} \)
17 \( 1 + (-0.0523 - 0.161i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-6.75 + 2.53i)T + (14.3 - 12.5i)T^{2} \)
23 \( 1 + (5.21 + 2.51i)T + (14.3 + 17.9i)T^{2} \)
29 \( 1 + (-4.33 - 0.194i)T + (28.8 + 2.59i)T^{2} \)
31 \( 1 + (1.13 - 1.90i)T + (-14.6 - 27.2i)T^{2} \)
37 \( 1 + (4.38 - 2.11i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + (-0.313 + 1.37i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (2.76 - 5.13i)T + (-23.6 - 35.8i)T^{2} \)
47 \( 1 + (5.31 - 8.04i)T + (-18.4 - 43.2i)T^{2} \)
53 \( 1 + (-4.02 - 1.11i)T + (45.4 + 27.1i)T^{2} \)
59 \( 1 + (-1.71 - 1.50i)T + (7.91 + 58.4i)T^{2} \)
61 \( 1 + (1.39 - 1.33i)T + (2.73 - 60.9i)T^{2} \)
67 \( 1 + (-2.04 - 7.40i)T + (-57.5 + 34.3i)T^{2} \)
73 \( 1 + (0.408 + 3.01i)T + (-70.3 + 19.4i)T^{2} \)
79 \( 1 + (6.59 + 15.4i)T + (-54.5 + 57.1i)T^{2} \)
83 \( 1 + (-7.50 + 8.59i)T + (-11.1 - 82.2i)T^{2} \)
89 \( 1 + (1.00 - 3.62i)T + (-76.4 - 45.6i)T^{2} \)
97 \( 1 + (2.16 - 0.494i)T + (87.3 - 42.0i)T^{2} \)
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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

−10.62843686059556606262775797649, −9.837219032334642002364761051510, −8.998884344427853326944160706393, −8.324171786972337781304667773024, −7.57623503326965999678124273324, −6.49727402072082750820394879959, −5.32580999575751549475976129646, −4.52211272720646495799289274143, −2.74232135271834350700373022311, −1.63569501069960912530896849072, 0.795830420464088503826949701128, 1.77641291565455885091973097064, 3.75224588484892137747696986973, 5.07294361366657254954754694856, 5.36814227342902962652953125995, 7.29692739285194696106299496362, 7.953668228140968363049079815267, 8.415282492808114265866137583800, 9.671652047091172144807817414633, 10.19010579373126548638422224145

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