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Properties

Label 2-63-63.58-c1-0-0
Degree $2$
Conductor $63$
Sign $0.524 - 0.851i$
Analytic cond. $0.503057$
Root an. cond. $0.709265$
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  − 2.38·2-s + (−1.61 − 0.624i)3-s + 3.69·4-s + (1.46 + 2.52i)5-s + (3.85 + 1.49i)6-s + (−0.138 + 2.64i)7-s − 4.05·8-s + (2.22 + 2.01i)9-s + (−3.48 − 6.03i)10-s + (0.676 − 1.17i)11-s + (−5.97 − 2.30i)12-s + (−0.733 + 1.26i)13-s + (0.330 − 6.30i)14-s + (−0.779 − 4.99i)15-s + 2.27·16-s + (1.65 + 2.86i)17-s + ⋯
L(s)  = 1  − 1.68·2-s + (−0.932 − 0.360i)3-s + 1.84·4-s + (0.653 + 1.13i)5-s + (1.57 + 0.608i)6-s + (−0.0523 + 0.998i)7-s − 1.43·8-s + (0.740 + 0.672i)9-s + (−1.10 − 1.90i)10-s + (0.204 − 0.353i)11-s + (−1.72 − 0.666i)12-s + (−0.203 + 0.352i)13-s + (0.0883 − 1.68i)14-s + (−0.201 − 1.29i)15-s + 0.568·16-s + (0.401 + 0.695i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.524 - 0.851i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ 0.524 - 0.851i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.310578 + 0.173449i\)
\(L(\frac12)\) \(\approx\) \(0.310578 + 0.173449i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.61 + 0.624i)T \)
7 \( 1 + (0.138 - 2.64i)T \)
good2 \( 1 + 2.38T + 2T^{2} \)
5 \( 1 + (-1.46 - 2.52i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.676 + 1.17i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.733 - 1.26i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.65 - 2.86i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.10 - 1.91i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.31 + 2.27i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.521 - 0.903i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.27T + 31T^{2} \)
37 \( 1 + (-5.43 + 9.41i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.904 - 1.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.17 + 3.76i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.97T + 47T^{2} \)
53 \( 1 + (3.22 + 5.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 - 0.559T + 61T^{2} \)
67 \( 1 - 12.8T + 67T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + (-5.22 - 9.05i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 0.767T + 79T^{2} \)
83 \( 1 + (0.983 + 1.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.20 + 5.54i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.14 + 7.17i)T + (-48.5 + 84.0i)T^{2} \)
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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

−15.58856527940623628035201096869, −14.27944654688132241796148986113, −12.47984237451165769061111487264, −11.33370516650147767858426348389, −10.50893846914160806738614822997, −9.579346621903137151468808291829, −8.146539393692235838238829035462, −6.76323029692715788084911017726, −5.95279916044332521466658862994, −2.15573284589757776053100995816, 1.04350541068247671363544768239, 4.82158448379452796949920558183, 6.53207955415737755395834729244, 7.85024191693842950769154239420, 9.393043253290350663579209940654, 9.914454380345203183807013819605, 11.00493970703555146550094405353, 12.19278269965763883612407905899, 13.50972962432257886535935106925, 15.48751780804510744008927440463

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