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Properties

Degree $2$
Conductor $63$
Sign $-0.502 - 0.864i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.02 + 3.51i)2-s + (−4.22 + 7.31i)4-s + (4.96 + 8.59i)5-s + (−15.3 + 10.2i)7-s − 1.80·8-s + (−20.1 + 34.8i)10-s + (6.76 − 11.7i)11-s + 18.5·13-s + (−67.3 − 33.1i)14-s + (30.1 + 52.1i)16-s + (46.8 − 81.1i)17-s + (−65.9 − 114. i)19-s − 83.7·20-s + 54.9·22-s + (99.1 + 171. i)23-s + ⋯
L(s)  = 1  + (0.716 + 1.24i)2-s + (−0.527 + 0.914i)4-s + (0.443 + 0.768i)5-s + (−0.831 + 0.556i)7-s − 0.0799·8-s + (−0.636 + 1.10i)10-s + (0.185 − 0.321i)11-s + 0.395·13-s + (−1.28 − 0.633i)14-s + (0.470 + 0.815i)16-s + (0.668 − 1.15i)17-s + (−0.796 − 1.37i)19-s − 0.936·20-s + 0.532·22-s + (0.898 + 1.55i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.502 - 0.864i$
Motivic weight: \(3\)
Character: $\chi_{63} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.502 - 0.864i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.02529 + 1.78143i\)
\(L(\frac12)\) \(\approx\) \(1.02529 + 1.78143i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (15.3 - 10.2i)T \)
good2 \( 1 + (-2.02 - 3.51i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-4.96 - 8.59i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-6.76 + 11.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 18.5T + 2.19e3T^{2} \)
17 \( 1 + (-46.8 + 81.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (65.9 + 114. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-99.1 - 171. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 188.T + 2.43e4T^{2} \)
31 \( 1 + (-41.9 + 72.6i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (40.0 + 69.4i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 385.T + 6.89e4T^{2} \)
43 \( 1 + 397.T + 7.95e4T^{2} \)
47 \( 1 + (136. + 235. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-18.4 + 32.0i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (197. - 342. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (6.73 + 11.6i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (170. - 294. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 211.T + 3.57e5T^{2} \)
73 \( 1 + (243. - 420. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (146. + 254. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 889.T + 5.71e5T^{2} \)
89 \( 1 + (572. + 991. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.38e3T + 9.12e5T^{2} \)
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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

−14.98351696631815818960060423461, −13.78822624368919646808531341619, −13.22563713432441441032903672113, −11.63150690160803745254525368244, −10.15506633700472494007720400720, −8.775629476317752737622205891348, −7.08902876729089466053736690147, −6.35651953521431817178971895270, −5.13935554047640458773678096032, −3.12677133772204655938795484185, 1.42920393168876694624862623410, 3.41440260665569850967033651579, 4.72268867019042596916604288154, 6.39273794203873742440701597406, 8.428660021837601699590444783422, 9.995009740440618997308902656475, 10.60553367017135951606488533254, 12.28669141882044647671550541861, 12.72944165625947850075368996642, 13.65883000194624079207461486149

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