↑

Properties

Label 2-63-9.4-c3-0-14
Degree $2$
Conductor $63$
Sign $0.948 + 0.318i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.66 + 2.88i)2-s + (−2.65 − 4.46i)3-s + (−1.55 + 2.68i)4-s + (8.37 − 14.5i)5-s + (8.46 − 15.1i)6-s + (−3.5 − 6.06i)7-s + 16.3·8-s + (−12.9 + 23.7i)9-s + 55.8·10-s + (4.84 + 8.39i)11-s + (16.1 − 0.205i)12-s + (7.83 − 13.5i)13-s + (11.6 − 20.2i)14-s + (−87.0 + 1.10i)15-s + (39.5 + 68.5i)16-s − 104.·17-s + ⋯
L(s)  = 1  + (0.589 + 1.02i)2-s + (−0.510 − 0.859i)3-s + (−0.194 + 0.336i)4-s + (0.748 − 1.29i)5-s + (0.576 − 1.02i)6-s + (−0.188 − 0.327i)7-s + 0.721·8-s + (−0.477 + 0.878i)9-s + 1.76·10-s + (0.132 + 0.230i)11-s + (0.387 − 0.00493i)12-s + (0.167 − 0.289i)13-s + (0.222 − 0.385i)14-s + (−1.49 + 0.0190i)15-s + (0.618 + 1.07i)16-s − 1.48·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.318i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.948 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.948 + 0.318i$
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: \((2,\ 63,\ (\ :3/2),\ 0.948 + 0.318i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.83046 - 0.298812i\)
\(L(\frac12)\) \(\approx\) \(1.83046 - 0.298812i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.65 + 4.46i)T \)
7 \( 1 + (3.5 + 6.06i)T \)
good2 \( 1 + (-1.66 - 2.88i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-8.37 + 14.5i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-4.84 - 8.39i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-7.83 + 13.5i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 104.T + 4.91e3T^{2} \)
19 \( 1 - 162.T + 6.85e3T^{2} \)
23 \( 1 + (36.0 - 62.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (32.8 + 56.9i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (98.8 - 171. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 123.T + 5.06e4T^{2} \)
41 \( 1 + (202. - 350. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-115. - 200. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-101. - 176. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 137.T + 1.48e5T^{2} \)
59 \( 1 + (-390. + 675. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (37.6 + 65.1i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-67.9 + 117. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 226.T + 3.57e5T^{2} \)
73 \( 1 - 148.T + 3.89e5T^{2} \)
79 \( 1 + (-459. - 796. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (482. + 836. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.25e3T + 7.04e5T^{2} \)
97 \( 1 + (415. + 719. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.07308424820834233729614707909, −13.39423230722067466121910289400, −12.71535336973830845497603681020, −11.26600215887489964292301400009, −9.599361867188093648396416718539, −8.070051624466392124392672955871, −6.85754696885102363112808351860, −5.71375048752401418980208659791, −4.81965162858817607578739051806, −1.37792147784222754215844223759, 2.54302131839918710492957752909, 3.83091700397251616567065446812, 5.53157198591707498061979206595, 6.90825412148029502233647705051, 9.297281565480350106568187443708, 10.35073777080724269401435722276, 11.13621071355370866425110762172, 11.92469040557183450534648962066, 13.45782779304612110978542769186, 14.27747137865901859815844503129

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