Properties

Degree $2$
Conductor $63$
Sign $0.699 + 0.714i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.91 − 2.26i)2-s + (6.22 + 10.7i)4-s + (−0.632 + 1.09i)5-s + (13.4 + 12.7i)7-s − 20.1i·8-s + (4.95 − 2.86i)10-s + (36.0 − 20.7i)11-s − 85.7i·13-s + (−23.5 − 80.3i)14-s + (4.27 − 7.39i)16-s + (38.8 + 67.3i)17-s + (42.1 + 24.3i)19-s − 15.7·20-s − 188.·22-s + (78.7 + 45.4i)23-s + ⋯
L(s)  = 1  + (−1.38 − 0.799i)2-s + (0.778 + 1.34i)4-s + (−0.0566 + 0.0980i)5-s + (0.723 + 0.690i)7-s − 0.890i·8-s + (0.156 − 0.0905i)10-s + (0.987 − 0.570i)11-s − 1.82i·13-s + (−0.450 − 1.53i)14-s + (0.0667 − 0.115i)16-s + (0.554 + 0.961i)17-s + (0.509 + 0.293i)19-s − 0.176·20-s − 1.82·22-s + (0.714 + 0.412i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.699 + 0.714i$
Motivic weight: \(3\)
Character: $\chi_{63} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.699 + 0.714i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.747394 - 0.314456i\)
\(L(\frac12)\) \(\approx\) \(0.747394 - 0.314456i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-13.4 - 12.7i)T \)
good2 \( 1 + (3.91 + 2.26i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (0.632 - 1.09i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-36.0 + 20.7i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 85.7iT - 2.19e3T^{2} \)
17 \( 1 + (-38.8 - 67.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-42.1 - 24.3i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-78.7 - 45.4i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 151. iT - 2.43e4T^{2} \)
31 \( 1 + (-76.3 + 44.0i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (45.2 - 78.4i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 383.T + 6.89e4T^{2} \)
43 \( 1 + 227.T + 7.95e4T^{2} \)
47 \( 1 + (-69.5 + 120. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (289. - 167. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (440. + 762. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-11.3 - 6.57i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-221. - 383. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 341. iT - 3.57e5T^{2} \)
73 \( 1 + (798. - 460. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-206. + 357. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 954.T + 5.71e5T^{2} \)
89 \( 1 + (14.8 - 25.7i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.19e3iT - 9.12e5T^{2} \)
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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

−14.49364755915064652165404426595, −12.76021877123707081266714039868, −11.66584243046835350857020383394, −10.85683723252129495286708315819, −9.710314803009756629872506896775, −8.547776248745503353974278325869, −7.76116356394393634572643608719, −5.66093747188360775098050894200, −3.12619248864572356698416935020, −1.21399393135536948119625264470, 1.27174938885264029213036465338, 4.54737115793970811358424949535, 6.68946453161276877716330390651, 7.38465215522889627635485943923, 8.819739819088519604687811031103, 9.573870359942017175765578853076, 10.91585594364515820606338798535, 12.04800420722256804578106093547, 14.02416935252430875096656222657, 14.68956978103512809103997358157

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