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Properties

Label 2-63-7.3-c6-0-11
Degree $2$
Conductor $63$
Sign $0.992 + 0.126i$
Analytic cond. $14.4934$
Root an. cond. $3.80702$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.76 + 4.78i)2-s + (16.7 − 28.9i)4-s + (57.9 − 33.4i)5-s + (−240. + 244. i)7-s + 538.·8-s + (320. + 185. i)10-s + (862. − 1.49e3i)11-s − 2.80e3i·13-s + (−1.83e3 − 475. i)14-s + (418. + 724. i)16-s + (5.32e3 + 3.07e3i)17-s + (7.73e3 − 4.46e3i)19-s − 2.24e3i·20-s + 9.53e3·22-s + (4.95e3 + 8.57e3i)23-s + ⋯
L(s)  = 1  + (0.345 + 0.598i)2-s + (0.261 − 0.452i)4-s + (0.463 − 0.267i)5-s + (−0.701 + 0.713i)7-s + 1.05·8-s + (0.320 + 0.185i)10-s + (0.648 − 1.12i)11-s − 1.27i·13-s + (−0.668 − 0.173i)14-s + (0.102 + 0.176i)16-s + (1.08 + 0.625i)17-s + (1.12 − 0.651i)19-s − 0.280i·20-s + 0.895·22-s + (0.406 + 0.704i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.126i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.992 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.992 + 0.126i$
Analytic conductor: \(14.4934\)
Root analytic conductor: \(3.80702\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3),\ 0.992 + 0.126i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.61297 - 0.165502i\)
\(L(\frac12)\) \(\approx\) \(2.61297 - 0.165502i\)
\(L(4)\) 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 \( 1 + (240. - 244. i)T \)
good2 \( 1 + (-2.76 - 4.78i)T + (-32 + 55.4i)T^{2} \)
5 \( 1 + (-57.9 + 33.4i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (-862. + 1.49e3i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 + 2.80e3iT - 4.82e6T^{2} \)
17 \( 1 + (-5.32e3 - 3.07e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-7.73e3 + 4.46e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (-4.95e3 - 8.57e3i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 - 1.36e4T + 5.94e8T^{2} \)
31 \( 1 + (2.18e4 + 1.26e4i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (1.13e4 + 1.96e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + 3.78e4iT - 4.75e9T^{2} \)
43 \( 1 - 7.36e4T + 6.32e9T^{2} \)
47 \( 1 + (1.20e5 - 6.95e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (8.25e3 - 1.43e4i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (-6.19e4 - 3.57e4i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (2.28e5 - 1.31e5i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (2.74e5 - 4.75e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + 4.65e5T + 1.28e11T^{2} \)
73 \( 1 + (2.89e5 + 1.66e5i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (-2.05e5 - 3.55e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 - 7.19e4iT - 3.26e11T^{2} \)
89 \( 1 + (-1.41e5 + 8.16e4i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 + 6.51e5iT - 8.32e11T^{2} \)
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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

−13.79189391060365340473476955574, −12.88217689820018594574943510498, −11.47236405202082236940047334740, −10.16788087251656740690822443961, −9.063321010018748909135176784330, −7.50973496874712229434302646841, −5.92314924547925952972175526661, −5.50721049272415562628027025024, −3.20818868486663485054658514777, −1.11126042508092688136442943578, 1.60241484255832131452342435639, 3.20540510577608200615066048016, 4.49272122013782799553849069733, 6.60729096871931194085546932845, 7.49814837953824909433843358328, 9.488946249391863818904555804389, 10.32271607885381108796051756286, 11.75476903725667613649523621133, 12.45196552307732983819474970651, 13.73341657248302055730314649952

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