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Properties

Label 2-63-9.2-c2-0-3
Degree $2$
Conductor $63$
Sign $0.162 - 0.986i$
Analytic cond. $1.71662$
Root an. cond. $1.31020$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.65 + 1.53i)2-s + (−2.10 + 2.13i)3-s + (2.70 + 4.67i)4-s + (0.225 − 0.130i)5-s + (−8.87 + 2.42i)6-s + (1.32 − 2.29i)7-s + 4.29i·8-s + (−0.0989 − 8.99i)9-s + 0.799·10-s + (11.1 + 6.43i)11-s + (−15.6 − 4.10i)12-s + (−1.86 − 3.23i)13-s + (7.02 − 4.05i)14-s + (−0.198 + 0.756i)15-s + (4.21 − 7.30i)16-s − 9.17i·17-s + ⋯
L(s)  = 1  + (1.32 + 0.766i)2-s + (−0.703 + 0.710i)3-s + (0.675 + 1.16i)4-s + (0.0451 − 0.0260i)5-s + (−1.47 + 0.404i)6-s + (0.188 − 0.327i)7-s + 0.536i·8-s + (−0.0109 − 0.999i)9-s + 0.0799·10-s + (1.01 + 0.585i)11-s + (−1.30 − 0.342i)12-s + (−0.143 − 0.248i)13-s + (0.501 − 0.289i)14-s + (−0.0132 + 0.0504i)15-s + (0.263 − 0.456i)16-s − 0.539i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.162 - 0.986i$
Analytic conductor: \(1.71662\)
Root analytic conductor: \(1.31020\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1),\ 0.162 - 0.986i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.43614 + 1.21859i\)
\(L(\frac12)\) \(\approx\) \(1.43614 + 1.21859i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.10 - 2.13i)T \)
7 \( 1 + (-1.32 + 2.29i)T \)
good2 \( 1 + (-2.65 - 1.53i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (-0.225 + 0.130i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-11.1 - 6.43i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (1.86 + 3.23i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 9.17iT - 289T^{2} \)
19 \( 1 + 33.2T + 361T^{2} \)
23 \( 1 + (-10.5 + 6.11i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (31.6 + 18.2i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-13.8 - 23.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 62.8T + 1.36e3T^{2} \)
41 \( 1 + (-31.1 + 17.9i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (8.88 - 15.3i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-58.2 - 33.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 72.9iT - 2.80e3T^{2} \)
59 \( 1 + (19.7 - 11.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-42.3 + 73.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-2.79 - 4.83i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 30.5iT - 5.04e3T^{2} \)
73 \( 1 + 39.4T + 5.32e3T^{2} \)
79 \( 1 + (-26.1 + 45.3i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-75.6 - 43.6i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 156. iT - 7.92e3T^{2} \)
97 \( 1 + (53.6 - 92.9i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−15.00015957346974022200709058672, −14.14402462515109751379415183263, −12.81581292127812193192151827818, −11.92468289100424477421640727812, −10.64592671218706582004901934506, −9.227537656171748355507650876754, −7.17760303744665112240327105964, −6.12421578028612616049723021780, −4.83516864446230445130880429075, −3.85980321311123811307107933781, 2.00488159895336805106268986793, 4.08628797776633372799204289686, 5.58307728896023995861149732757, 6.59514972573621081974416777566, 8.525690708235724237645267611913, 10.59571567509683679293116160792, 11.48728741259494813554732976385, 12.28547222937960571058316069111, 13.15679467731636488716069014914, 14.13788361610586496448533004647

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