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Properties

Label 2-63-21.17-c5-0-11
Degree $2$
Conductor $63$
Sign $0.174 + 0.984i$
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  + (9.49 − 5.47i)2-s + (44.0 − 76.2i)4-s + (41.3 + 71.5i)5-s + (−18.0 − 128. i)7-s − 614. i·8-s + (784. + 452. i)10-s + (98.5 + 56.9i)11-s + 329. i·13-s + (−875. − 1.11e3i)14-s + (−1.95e3 − 3.39e3i)16-s + (−221. + 383. i)17-s + (−1.45e3 + 839. i)19-s + 7.27e3·20-s + 1.24e3·22-s + (−194. + 112. i)23-s + ⋯
L(s)  = 1  + (1.67 − 0.968i)2-s + (1.37 − 2.38i)4-s + (0.738 + 1.27i)5-s + (−0.139 − 0.990i)7-s − 3.39i·8-s + (2.47 + 1.43i)10-s + (0.245 + 0.141i)11-s + 0.540i·13-s + (−1.19 − 1.52i)14-s + (−1.91 − 3.31i)16-s + (−0.185 + 0.322i)17-s + (−0.924 + 0.533i)19-s + 4.06·20-s + 0.549·22-s + (−0.0764 + 0.0441i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.174 + 0.984i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.174 + 0.984i$
Analytic conductor: \(10.1041\)
Root analytic conductor: \(3.17870\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :5/2),\ 0.174 + 0.984i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.56295 - 2.98845i\)
\(L(\frac12)\) \(\approx\) \(3.56295 - 2.98845i\)
\(L(\frac{7}{2})\) 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 \( 1 + (18.0 + 128. i)T \)
good2 \( 1 + (-9.49 + 5.47i)T + (16 - 27.7i)T^{2} \)
5 \( 1 + (-41.3 - 71.5i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-98.5 - 56.9i)T + (8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 329. iT - 3.71e5T^{2} \)
17 \( 1 + (221. - 383. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (1.45e3 - 839. i)T + (1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (194. - 112. i)T + (3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 2.98e3iT - 2.05e7T^{2} \)
31 \( 1 + (-2.92e3 - 1.68e3i)T + (1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-5.85e3 - 1.01e4i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 1.45e4T + 1.15e8T^{2} \)
43 \( 1 + 1.05e4T + 1.47e8T^{2} \)
47 \( 1 + (1.10e4 + 1.91e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-3.94e3 - 2.27e3i)T + (2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-6.66e3 + 1.15e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (4.39e4 - 2.53e4i)T + (4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-2.56e4 + 4.44e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 1.20e3iT - 1.80e9T^{2} \)
73 \( 1 + (-7.65e3 - 4.41e3i)T + (1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (2.17e4 + 3.77e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 5.81e4T + 3.93e9T^{2} \)
89 \( 1 + (5.42e4 + 9.39e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 5.44e3iT - 8.58e9T^{2} \)
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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

−13.77080057862074486533884352426, −12.82751680742636039246374569823, −11.47530362020447928152344783150, −10.57041522067587065686977069771, −9.935019249248231758804614413406, −6.87603565931917507653430157103, −6.16195292964964203691002078592, −4.41105969383153940299169222093, −3.17376334108011315332634918237, −1.76104400764586708666689105238, 2.49361302870476129363484033348, 4.43125205252240557024698695146, 5.48823898064678169269502948499, 6.30508190888921124937273292576, 8.068682663443561878889117333783, 9.147808987510799945754247214497, 11.48433787534546354063180759545, 12.67170387960636129265428812307, 13.00893295403762841283067124984, 14.16135158189516614081690717581

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