Properties

Degree $2$
Conductor $63$
Sign $-0.645 - 0.763i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.96 + 3.40i)2-s + (1.35 + 5.01i)3-s + (−3.73 + 6.47i)4-s + (1.21 − 2.10i)5-s + (−14.4 + 14.4i)6-s + (3.5 + 6.06i)7-s + 2.05·8-s + (−23.3 + 13.5i)9-s + 9.56·10-s + (−21.0 − 36.5i)11-s + (−37.5 − 9.98i)12-s + (19.7 − 34.2i)13-s + (−13.7 + 23.8i)14-s + (12.2 + 3.24i)15-s + (33.9 + 58.8i)16-s − 2.07·17-s + ⋯
L(s)  = 1  + (0.695 + 1.20i)2-s + (0.260 + 0.965i)3-s + (−0.467 + 0.809i)4-s + (0.108 − 0.188i)5-s + (−0.981 + 0.985i)6-s + (0.188 + 0.327i)7-s + 0.0908·8-s + (−0.864 + 0.503i)9-s + 0.302·10-s + (−0.577 − 1.00i)11-s + (−0.903 − 0.240i)12-s + (0.422 − 0.731i)13-s + (−0.262 + 0.455i)14-s + (0.210 + 0.0559i)15-s + (0.530 + 0.918i)16-s − 0.0296·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.645 - 0.763i$
Motivic weight: \(3\)
Character: $\chi_{63} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.645 - 0.763i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.921949 + 1.98659i\)
\(L(\frac12)\) \(\approx\) \(0.921949 + 1.98659i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.35 - 5.01i)T \)
7 \( 1 + (-3.5 - 6.06i)T \)
good2 \( 1 + (-1.96 - 3.40i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-1.21 + 2.10i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (21.0 + 36.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-19.7 + 34.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 2.07T + 4.91e3T^{2} \)
19 \( 1 - 96.1T + 6.85e3T^{2} \)
23 \( 1 + (36.8 - 63.8i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (9.60 + 16.6i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-119. + 207. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 144.T + 5.06e4T^{2} \)
41 \( 1 + (-36.1 + 62.5i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (240. + 416. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (147. + 255. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 627.T + 1.48e5T^{2} \)
59 \( 1 + (74.8 - 129. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-315. - 546. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (2.02 - 3.50i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 798.T + 3.57e5T^{2} \)
73 \( 1 - 444.T + 3.89e5T^{2} \)
79 \( 1 + (-287. - 498. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (645. + 1.11e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 750.T + 7.04e5T^{2} \)
97 \( 1 + (209. + 363. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−15.12455060735536216620841055418, −13.93714222242288141104526336229, −13.28567523055634751552037887026, −11.44189433514867506356987495250, −10.22236420588122233638198079703, −8.722393971435654499743446142510, −7.72698883461432979540651918748, −5.82075232505553734875261904735, −5.12950237626182207711121466011, −3.45121627763063097762332086733, 1.57228710194750973228350208005, 3.00260165427286123950953129186, 4.76330297943681488835286379363, 6.72711801267626040141738971054, 8.000618949691579433769865046216, 9.761855949980013993340333798296, 11.00443418603184938114532942087, 12.04637337460340062231830588914, 12.79799601668443518099182291297, 13.84538192132800832361311650073

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