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Properties

Label 2-63-7.4-c5-0-3
Degree $2$
Conductor $63$
Sign $0.962 - 0.273i$
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  + (−2.44 − 4.23i)2-s + (4.04 − 7.00i)4-s + (21.3 + 37.0i)5-s + (43.2 + 122. i)7-s − 196.·8-s + (104. − 181. i)10-s + (−355. + 615. i)11-s + 885.·13-s + (411. − 481. i)14-s + (349. + 605. i)16-s + (350. − 607. i)17-s + (627. + 1.08e3i)19-s + 345.·20-s + 3.47e3·22-s + (523. + 906. i)23-s + ⋯
L(s)  = 1  + (−0.432 − 0.748i)2-s + (0.126 − 0.219i)4-s + (0.382 + 0.662i)5-s + (0.333 + 0.942i)7-s − 1.08·8-s + (0.330 − 0.572i)10-s + (−0.885 + 1.53i)11-s + 1.45·13-s + (0.561 − 0.657i)14-s + (0.341 + 0.591i)16-s + (0.294 − 0.509i)17-s + (0.399 + 0.691i)19-s + 0.193·20-s + 1.53·22-s + (0.206 + 0.357i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.273i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.962 - 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.962 - 0.273i$
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: \((2,\ 63,\ (\ :5/2),\ 0.962 - 0.273i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.41762 + 0.197285i\)
\(L(\frac12)\) \(\approx\) \(1.41762 + 0.197285i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-43.2 - 122. i)T \)
good2 \( 1 + (2.44 + 4.23i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (-21.3 - 37.0i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (355. - 615. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 885.T + 3.71e5T^{2} \)
17 \( 1 + (-350. + 607. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-627. - 1.08e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-523. - 906. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 6.15e3T + 2.05e7T^{2} \)
31 \( 1 + (1.14e3 - 1.98e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (202. + 349. i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.78e4T + 1.15e8T^{2} \)
43 \( 1 + 1.46e4T + 1.47e8T^{2} \)
47 \( 1 + (-1.05e4 - 1.83e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (53.6 - 92.8i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-2.22e4 + 3.85e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.16e4 - 2.01e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-3.33e3 + 5.77e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 2.51e4T + 1.80e9T^{2} \)
73 \( 1 + (4.73e3 - 8.20e3i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-1.32e4 - 2.29e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 7.49e3T + 3.93e9T^{2} \)
89 \( 1 + (1.60e4 + 2.78e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.55e5T + 8.58e9T^{2} \)
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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

−14.09149093873541031598996588206, −12.57178759089621464448134635304, −11.61593991443827032408818663063, −10.48874096009288516450101096682, −9.732017246765153082062072759509, −8.372002678532643089343562625063, −6.62862699214713072443855765288, −5.31868938362709709086369339041, −2.89974664570084167583647620332, −1.66693094890332641188782487930, 0.815280316097173741732453913993, 3.40359457452569665556106217238, 5.42317196018258466553962216442, 6.68885898116134413100007129236, 8.189736183154214346261771431426, 8.703502840673316166858228875972, 10.45829355915656376566771111354, 11.53317165782000312662677720281, 13.16354572872573909828543416782, 13.74768656303858319934077023083

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