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Properties

Label 2-63-63.47-c3-0-10
Degree $2$
Conductor $63$
Sign $0.946 - 0.324i$
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  + (−3.22 + 1.86i)2-s + (4.92 − 1.65i)3-s + (2.92 − 5.06i)4-s + 2.66·5-s + (−12.7 + 14.4i)6-s + (9.53 − 15.8i)7-s − 8.01i·8-s + (21.5 − 16.3i)9-s + (−8.57 + 4.94i)10-s + 67.9i·11-s + (6.01 − 29.7i)12-s + (69.6 − 40.2i)13-s + (−1.16 + 68.9i)14-s + (13.1 − 4.40i)15-s + (38.2 + 66.3i)16-s + (29.6 + 51.2i)17-s + ⋯
L(s)  = 1  + (−1.13 + 0.657i)2-s + (0.947 − 0.318i)3-s + (0.365 − 0.632i)4-s + 0.237·5-s + (−0.870 + 0.986i)6-s + (0.514 − 0.857i)7-s − 0.354i·8-s + (0.797 − 0.603i)9-s + (−0.271 + 0.156i)10-s + 1.86i·11-s + (0.144 − 0.716i)12-s + (1.48 − 0.858i)13-s + (−0.0223 + 1.31i)14-s + (0.225 − 0.0757i)15-s + (0.598 + 1.03i)16-s + (0.422 + 0.731i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.324i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.946 - 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.946 - 0.324i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.946 - 0.324i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.18961 + 0.198140i\)
\(L(\frac12)\) \(\approx\) \(1.18961 + 0.198140i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.92 + 1.65i)T \)
7 \( 1 + (-9.53 + 15.8i)T \)
good2 \( 1 + (3.22 - 1.86i)T + (4 - 6.92i)T^{2} \)
5 \( 1 - 2.66T + 125T^{2} \)
11 \( 1 - 67.9iT - 1.33e3T^{2} \)
13 \( 1 + (-69.6 + 40.2i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-29.6 - 51.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (5.86 + 3.38i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 126. iT - 1.21e4T^{2} \)
29 \( 1 + (114. + 66.0i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-62.8 - 36.2i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (63.8 - 110. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-4.93 - 8.54i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (108. - 188. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (92.9 + 160. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (174. - 100. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (89.2 - 154. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (195. - 112. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (219. - 379. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 533. iT - 3.57e5T^{2} \)
73 \( 1 + (-287. + 165. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-503. - 872. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (590. - 1.02e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-66.3 + 114. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (761. + 439. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−14.84958351289511377874195841518, −13.49587380612543287124042395428, −12.58841629912164209203659870348, −10.47980296517647543594109882244, −9.711704164786437709198575738535, −8.360638508147312743773051706998, −7.69044506218309251352651638504, −6.56002258649840268273242160910, −4.01791789316479961358183925970, −1.46431241551873524859177441055, 1.65042275499906889471913869565, 3.31473002751837786329521889983, 5.66331225473738049919861984425, 7.986047356476503199087887721812, 8.804358273807793956004394195117, 9.444108583119983505625298240729, 10.93363436818423673926474424304, 11.59849587776827833991541164460, 13.58778099057823970714468979402, 14.19552931563778685540083441085

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