Properties

Label 2-63-9.7-c3-0-10
Degree $2$
Conductor $63$
Sign $0.961 + 0.273i$
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  + (1.46 − 2.54i)2-s + (3.48 + 3.85i)3-s + (−0.320 − 0.554i)4-s + (1.28 + 2.22i)5-s + (14.9 − 3.19i)6-s + (3.5 − 6.06i)7-s + 21.6·8-s + (−2.75 + 26.8i)9-s + 7.55·10-s + (0.257 − 0.445i)11-s + (1.02 − 3.16i)12-s + (−32.7 − 56.6i)13-s + (−10.2 − 17.8i)14-s + (−4.10 + 12.6i)15-s + (34.3 − 59.5i)16-s − 3.37·17-s + ⋯
L(s)  = 1  + (0.519 − 0.900i)2-s + (0.670 + 0.742i)3-s + (−0.0400 − 0.0693i)4-s + (0.114 + 0.198i)5-s + (1.01 − 0.217i)6-s + (0.188 − 0.327i)7-s + 0.956·8-s + (−0.102 + 0.994i)9-s + 0.238·10-s + (0.00704 − 0.0122i)11-s + (0.0246 − 0.0761i)12-s + (−0.698 − 1.20i)13-s + (−0.196 − 0.340i)14-s + (−0.0707 + 0.218i)15-s + (0.536 − 0.929i)16-s − 0.0481·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.273i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.961 + 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.961 + 0.273i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.961 + 0.273i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.32496 - 0.323956i\)
\(L(\frac12)\) \(\approx\) \(2.32496 - 0.323956i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.48 - 3.85i)T \)
7 \( 1 + (-3.5 + 6.06i)T \)
good2 \( 1 + (-1.46 + 2.54i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (-1.28 - 2.22i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-0.257 + 0.445i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (32.7 + 56.6i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 3.37T + 4.91e3T^{2} \)
19 \( 1 + 123.T + 6.85e3T^{2} \)
23 \( 1 + (-12.5 - 21.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (118. - 205. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-44.9 - 77.8i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 67.4T + 5.06e4T^{2} \)
41 \( 1 + (143. + 248. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-217. + 377. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (27.0 - 46.8i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 272.T + 1.48e5T^{2} \)
59 \( 1 + (-258. - 447. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (125. - 217. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (230. + 399. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 532.T + 3.57e5T^{2} \)
73 \( 1 + 360.T + 3.89e5T^{2} \)
79 \( 1 + (381. - 660. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-472. + 818. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.49e3T + 7.04e5T^{2} \)
97 \( 1 + (672. - 1.16e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−14.31290672070765176674296785963, −13.24554763852265321270286394089, −12.28823985851011750445903232069, −10.67058531111940896995043522782, −10.39416101615419235353268839226, −8.668555320545624689455506345357, −7.37884681820944062013427873795, −5.02774210890273958334350313178, −3.71430661312872125517713898861, −2.40098738804147897123503400960, 2.00542651801110699578233678277, 4.44643751615384925122696485679, 6.11353798426221569010324684675, 7.09810911494247843046942058653, 8.297313049394330431282256426763, 9.572498854180343933943158155041, 11.36608047455117486031151656693, 12.72488533401087084428874125943, 13.60969594238369902088688308511, 14.67259419806443913619190696775

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