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Properties

Label 2-63-7.4-c5-0-0
Degree $2$
Conductor $63$
Sign $0.779 - 0.626i$
Analytic cond. $10.1041$
Root an. cond. $3.17870$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.69 − 8.13i)2-s + (−28.0 + 48.6i)4-s + (−35.8 − 62.1i)5-s + (−87.5 − 95.6i)7-s + 226.·8-s + (−336. + 583. i)10-s + (−280. + 485. i)11-s + 533.·13-s + (−367. + 1.16e3i)14-s + (−166. − 288. i)16-s + (502. − 870. i)17-s + (−684. − 1.18e3i)19-s + 4.02e3·20-s + 5.26e3·22-s + (1.61e3 + 2.79e3i)23-s + ⋯
L(s)  = 1  + (−0.829 − 1.43i)2-s + (−0.877 + 1.52i)4-s + (−0.641 − 1.11i)5-s + (−0.674 − 0.737i)7-s + 1.25·8-s + (−1.06 + 1.84i)10-s + (−0.698 + 1.20i)11-s + 0.875·13-s + (−0.500 + 1.58i)14-s + (−0.162 − 0.282i)16-s + (0.422 − 0.730i)17-s + (−0.434 − 0.753i)19-s + 2.25·20-s + 2.31·22-s + (0.636 + 1.10i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.779 - 0.626i$
Analytic conductor: \(10.1041\)
Root analytic conductor: \(3.17870\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :5/2),\ 0.779 - 0.626i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0347749 + 0.0122523i\)
\(L(\frac12)\) \(\approx\) \(0.0347749 + 0.0122523i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (87.5 + 95.6i)T \)
good2 \( 1 + (4.69 + 8.13i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (35.8 + 62.1i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (280. - 485. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 533.T + 3.71e5T^{2} \)
17 \( 1 + (-502. + 870. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (684. + 1.18e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-1.61e3 - 2.79e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 753.T + 2.05e7T^{2} \)
31 \( 1 + (4.10e3 - 7.10e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-1.40e3 - 2.43e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 245.T + 1.15e8T^{2} \)
43 \( 1 + 1.75e4T + 1.47e8T^{2} \)
47 \( 1 + (8.17e3 + 1.41e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (1.48e4 - 2.56e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (5.17e3 - 8.96e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (477. + 826. i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-9.90e3 + 1.71e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 6.21e4T + 1.80e9T^{2} \)
73 \( 1 + (1.35e4 - 2.34e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (2.23e4 + 3.87e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 1.56e4T + 3.93e9T^{2} \)
89 \( 1 + (-6.81e3 - 1.18e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.29e4T + 8.58e9T^{2} \)
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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

−13.35829286628030071864186910496, −12.73503720025932366902429563953, −11.76458121194723360030616197019, −10.61475800952437390778390295780, −9.597209013969560943962924146111, −8.628038721161768062305420744942, −7.33281183791811329242235070304, −4.70761905813971803437664999013, −3.30057261407619183337907937098, −1.29859685960951699672023475200, 0.02495969997281049475921872076, 3.27151685567271191505838159799, 5.82038328662090562353090139701, 6.53325552458356396529214248068, 7.927480202357065839632003425544, 8.692869852495985342797730489107, 10.17693526802358018132177570504, 11.23933441528359290480065793697, 12.98269252664169310563241285790, 14.47776473903066857135689656979

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