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Properties

Label 2-63-9.4-c3-0-4
Degree $2$
Conductor $63$
Sign $-0.999 - 0.0322i$
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  + (2.21 + 3.83i)2-s + (−4.03 + 3.27i)3-s + (−5.78 + 10.0i)4-s + (−1.92 + 3.33i)5-s + (−21.4 − 8.21i)6-s + (−3.5 − 6.06i)7-s − 15.8·8-s + (5.54 − 26.4i)9-s − 17.0·10-s + (19.8 + 34.3i)11-s + (−9.49 − 59.4i)12-s + (−31.7 + 54.9i)13-s + (15.4 − 26.8i)14-s + (−3.15 − 19.7i)15-s + (11.2 + 19.5i)16-s + 10.5·17-s + ⋯
L(s)  = 1  + (0.782 + 1.35i)2-s + (−0.776 + 0.630i)3-s + (−0.723 + 1.25i)4-s + (−0.172 + 0.298i)5-s + (−1.46 − 0.558i)6-s + (−0.188 − 0.327i)7-s − 0.699·8-s + (0.205 − 0.978i)9-s − 0.539·10-s + (0.543 + 0.940i)11-s + (−0.228 − 1.42i)12-s + (−0.676 + 1.17i)13-s + (0.295 − 0.512i)14-s + (−0.0543 − 0.340i)15-s + (0.176 + 0.305i)16-s + 0.149·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0322i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.999 - 0.0322i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.999 - 0.0322i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0239200 + 1.48450i\)
\(L(\frac12)\) \(\approx\) \(0.0239200 + 1.48450i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.03 - 3.27i)T \)
7 \( 1 + (3.5 + 6.06i)T \)
good2 \( 1 + (-2.21 - 3.83i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (1.92 - 3.33i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-19.8 - 34.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (31.7 - 54.9i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 10.5T + 4.91e3T^{2} \)
19 \( 1 + 8.40T + 6.85e3T^{2} \)
23 \( 1 + (-87.2 + 151. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-42.0 - 72.8i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-11.4 + 19.8i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 302.T + 5.06e4T^{2} \)
41 \( 1 + (-179. + 310. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (170. + 296. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (153. + 265. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 192.T + 1.48e5T^{2} \)
59 \( 1 + (-151. + 262. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-204. - 354. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (196. - 339. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 901.T + 3.57e5T^{2} \)
73 \( 1 + 1.13e3T + 3.89e5T^{2} \)
79 \( 1 + (242. + 419. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (51.9 + 89.9i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.42e3T + 7.04e5T^{2} \)
97 \( 1 + (-752. - 1.30e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−14.84593976146715264814188616105, −14.52937520865019383782641354154, −12.94111575455830053035160818192, −11.87822129372079555234305671183, −10.47560190353840041275548924002, −9.117208486125087502303783603316, −7.15755300990802905667304498939, −6.56184242542049838914721577178, −4.99365224062631090307197668209, −4.06905089114275390273927867036, 0.962290245286800925704588716498, 2.94827469417589733591298550387, 4.81005545215532023449994125745, 6.00720627390864487433701243911, 7.893402339577384060157662161738, 9.759274811355854672196559301516, 11.03263979904749435814849058325, 11.73410486008097437641927727077, 12.71475318950362669182629677513, 13.30726314737213853623502599997

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