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Properties

Label 2-63-63.25-c1-0-1
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: $\chi_{63} (25, \cdot )$
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

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

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