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Properties

Label 2-63-63.58-c3-0-21
Degree $2$
Conductor $63$
Sign $-0.787 + 0.616i$
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  + 2.66·2-s + (−4.74 − 2.11i)3-s − 0.908·4-s + (−9.61 − 16.6i)5-s + (−12.6 − 5.64i)6-s + (−5.55 + 17.6i)7-s − 23.7·8-s + (18.0 + 20.1i)9-s + (−25.6 − 44.3i)10-s + (19.2 − 33.3i)11-s + (4.31 + 1.92i)12-s + (38.2 − 66.3i)13-s + (−14.7 + 47.0i)14-s + (10.3 + 99.4i)15-s − 55.9·16-s + (−11.9 − 20.6i)17-s + ⋯
L(s)  = 1  + 0.941·2-s + (−0.913 − 0.407i)3-s − 0.113·4-s + (−0.860 − 1.49i)5-s + (−0.859 − 0.383i)6-s + (−0.300 + 0.953i)7-s − 1.04·8-s + (0.667 + 0.744i)9-s + (−0.810 − 1.40i)10-s + (0.528 − 0.914i)11-s + (0.103 + 0.0463i)12-s + (0.816 − 1.41i)13-s + (−0.282 + 0.898i)14-s + (0.178 + 1.71i)15-s − 0.873·16-s + (−0.170 − 0.294i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.787 + 0.616i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.787 + 0.616i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.787 + 0.616i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.281111 - 0.815283i\)
\(L(\frac12)\) \(\approx\) \(0.281111 - 0.815283i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.74 + 2.11i)T \)
7 \( 1 + (5.55 - 17.6i)T \)
good2 \( 1 - 2.66T + 8T^{2} \)
5 \( 1 + (9.61 + 16.6i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-19.2 + 33.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-38.2 + 66.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (11.9 + 20.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (23.6 - 41.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-6.76 - 11.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (53.5 + 92.6i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 158.T + 2.97e4T^{2} \)
37 \( 1 + (-23.0 + 39.8i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-101. + 175. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (42.1 + 72.9i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 473.T + 1.03e5T^{2} \)
53 \( 1 + (39.8 + 69.0i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + 316.T + 2.05e5T^{2} \)
61 \( 1 + 163.T + 2.26e5T^{2} \)
67 \( 1 - 540.T + 3.00e5T^{2} \)
71 \( 1 - 810.T + 3.57e5T^{2} \)
73 \( 1 + (142. + 246. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 - 734.T + 4.93e5T^{2} \)
83 \( 1 + (290. + 503. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (463. - 803. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (413. + 715. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.57668875396018167545990205182, −12.69523227623165453049035003738, −12.23942895450341315947848191210, −11.20437692784119164812185996216, −9.100711478478165255807659356289, −8.137848398960726319852628031650, −5.93169854885986007901202847631, −5.29374978608166097347646730324, −3.82267800353735625044096171854, −0.50586285799659076219105463832, 3.71094067923724518689643037568, 4.36089151134756168437571066057, 6.40952314181533366139438375430, 7.05097000580162066216612535976, 9.413172622257618728187694495300, 10.79194909443337066795940493394, 11.47876747580132075788918093677, 12.61315562127005607184514667128, 13.97767391680761060875173877274, 14.80948214905121232208901944604

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