Properties

Degree $2$
Conductor $63$
Sign $0.337 - 0.941i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.797 + 1.38i)2-s + (5.17 + 0.441i)3-s + (2.72 + 4.72i)4-s + (1.27 + 2.21i)5-s + (−4.73 + 6.79i)6-s + (3.5 − 6.06i)7-s − 21.4·8-s + (26.6 + 4.57i)9-s − 4.07·10-s + (−4.04 + 7.00i)11-s + (12.0 + 25.6i)12-s + (13.1 + 22.7i)13-s + (5.58 + 9.66i)14-s + (5.64 + 12.0i)15-s + (−4.70 + 8.15i)16-s − 69.7·17-s + ⋯
L(s)  = 1  + (−0.281 + 0.488i)2-s + (0.996 + 0.0849i)3-s + (0.341 + 0.590i)4-s + (0.114 + 0.198i)5-s + (−0.322 + 0.462i)6-s + (0.188 − 0.327i)7-s − 0.948·8-s + (0.985 + 0.169i)9-s − 0.128·10-s + (−0.110 + 0.192i)11-s + (0.289 + 0.617i)12-s + (0.279 + 0.484i)13-s + (0.106 + 0.184i)14-s + (0.0970 + 0.207i)15-s + (−0.0735 + 0.127i)16-s − 0.994·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.337 - 0.941i$
Motivic weight: \(3\)
Character: $\chi_{63} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.337 - 0.941i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.47344 + 1.03660i\)
\(L(\frac12)\) \(\approx\) \(1.47344 + 1.03660i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.17 - 0.441i)T \)
7 \( 1 + (-3.5 + 6.06i)T \)
good2 \( 1 + (0.797 - 1.38i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (-1.27 - 2.21i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (4.04 - 7.00i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-13.1 - 22.7i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 69.7T + 4.91e3T^{2} \)
19 \( 1 - 105.T + 6.85e3T^{2} \)
23 \( 1 + (77.1 + 133. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-36.3 + 62.9i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (141. + 244. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 25.7T + 5.06e4T^{2} \)
41 \( 1 + (-43.5 - 75.4i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (44.5 - 77.1i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (157. - 272. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 356.T + 1.48e5T^{2} \)
59 \( 1 + (206. + 357. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (73.3 - 127. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-153. - 265. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 1.03e3T + 3.57e5T^{2} \)
73 \( 1 + 1.15e3T + 3.89e5T^{2} \)
79 \( 1 + (373. - 646. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (262. - 455. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 643.T + 7.04e5T^{2} \)
97 \( 1 + (154. - 267. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−14.75064441657799215541752067979, −13.76697841072065937199193354210, −12.63639672036886688891971180388, −11.26869975440104597201098480764, −9.768623711851717658042907096505, −8.608876302140135230489138222255, −7.63667021614699320675762186287, −6.54576521664258995036160714273, −4.15144209598724826996625175124, −2.51054392916014174353828992722, 1.58228085817417117343241623447, 3.18971955606463887201153533825, 5.42242914733977227882534506682, 7.14007766709216783136864703784, 8.664231892637612841027463308798, 9.528348648232274489073680922743, 10.69904346314718522141686078607, 11.92221027282640225169951838838, 13.23216491623154485684932476607, 14.24946860810749903190507117583

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