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Properties

Label 2-63-63.20-c3-0-4
Degree $2$
Conductor $63$
Sign $0.702 + 0.711i$
Analytic cond. $3.71712$
Root an. cond. $1.92798$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.67 − 2.70i)2-s + (−2.75 − 4.40i)3-s + (10.5 + 18.3i)4-s + (3.43 + 5.95i)5-s + (1.00 + 28.0i)6-s + (14.2 + 11.7i)7-s − 71.1i·8-s + (−11.8 + 24.2i)9-s − 37.1i·10-s + (−10.9 − 6.30i)11-s + (51.5 − 97.2i)12-s + (22.5 − 13.0i)13-s + (−35.0 − 93.6i)14-s + (16.7 − 31.5i)15-s + (−107. + 186. i)16-s + 124.·17-s + ⋯
L(s)  = 1  + (−1.65 − 0.954i)2-s + (−0.530 − 0.847i)3-s + (1.32 + 2.29i)4-s + (0.307 + 0.532i)5-s + (0.0680 + 1.90i)6-s + (0.771 + 0.635i)7-s − 3.14i·8-s + (−0.437 + 0.899i)9-s − 1.17i·10-s + (−0.299 − 0.172i)11-s + (1.24 − 2.33i)12-s + (0.481 − 0.277i)13-s + (−0.669 − 1.78i)14-s + (0.288 − 0.543i)15-s + (−1.68 + 2.91i)16-s + 1.77·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.702 + 0.711i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.702 + 0.711i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.579364 - 0.242288i\)
\(L(\frac12)\) \(\approx\) \(0.579364 - 0.242288i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.75 + 4.40i)T \)
7 \( 1 + (-14.2 - 11.7i)T \)
good2 \( 1 + (4.67 + 2.70i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (-3.43 - 5.95i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (10.9 + 6.30i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-22.5 + 13.0i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 124.T + 4.91e3T^{2} \)
19 \( 1 - 41.3iT - 6.85e3T^{2} \)
23 \( 1 + (97.6 - 56.4i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-114. - 66.3i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-155. + 89.5i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 148.T + 5.06e4T^{2} \)
41 \( 1 + (-93.6 - 162. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-99.1 + 171. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (92.1 - 159. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 359. iT - 1.48e5T^{2} \)
59 \( 1 + (182. + 315. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (300. + 173. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (182. + 315. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 565. iT - 3.57e5T^{2} \)
73 \( 1 - 737. iT - 3.89e5T^{2} \)
79 \( 1 + (451. - 782. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-382. + 662. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 395.T + 7.04e5T^{2} \)
97 \( 1 + (243. + 140. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−14.12996611559726415379527130084, −12.52902321305831942698884489663, −11.82232370503484632261594051968, −10.85850797325122479392061501294, −9.937531319399581332125529898431, −8.298716770629557680534785441323, −7.68938796149349572707212139352, −6.04006163685245305071547207271, −2.73502038637725222356455366792, −1.26650300347367577105393615816, 0.992991805492899770054351099122, 4.94986647347441675203696747337, 6.15689566587013144239098323383, 7.69711970645053735712314737506, 8.774495952629895449681044892533, 9.917305373359139066199788119685, 10.59544098074141550835967837902, 11.78989404680455318623859525632, 14.10850110783079445411399137959, 15.04367729546552131077923650995

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