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Properties

Label 2-63-63.38-c3-0-9
Degree $2$
Conductor $63$
Sign $0.854 + 0.519i$
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  − 1.15i·2-s + (−5.00 + 1.39i)3-s + 6.67·4-s + (0.137 − 0.237i)5-s + (1.60 + 5.77i)6-s + (18.5 − 0.122i)7-s − 16.9i·8-s + (23.1 − 13.9i)9-s + (−0.274 − 0.158i)10-s + (22.7 − 13.1i)11-s + (−33.4 + 9.27i)12-s + (39.6 − 22.8i)13-s + (−0.141 − 21.3i)14-s + (−0.356 + 1.38i)15-s + 33.8·16-s + (−30.2 + 52.3i)17-s + ⋯
L(s)  = 1  − 0.407i·2-s + (−0.963 + 0.267i)3-s + 0.833·4-s + (0.0122 − 0.0212i)5-s + (0.109 + 0.392i)6-s + (0.999 − 0.00663i)7-s − 0.747i·8-s + (0.856 − 0.515i)9-s + (−0.00867 − 0.00500i)10-s + (0.622 − 0.359i)11-s + (−0.803 + 0.223i)12-s + (0.845 − 0.487i)13-s + (−0.00270 − 0.407i)14-s + (−0.00614 + 0.0237i)15-s + 0.529·16-s + (−0.431 + 0.747i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.854 + 0.519i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.854 + 0.519i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.39721 - 0.391624i\)
\(L(\frac12)\) \(\approx\) \(1.39721 - 0.391624i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.00 - 1.39i)T \)
7 \( 1 + (-18.5 + 0.122i)T \)
good2 \( 1 + 1.15iT - 8T^{2} \)
5 \( 1 + (-0.137 + 0.237i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-22.7 + 13.1i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-39.6 + 22.8i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (30.2 - 52.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (38.8 - 22.4i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (110. + 63.8i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-58.2 - 33.6i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 98.9iT - 2.97e4T^{2} \)
37 \( 1 + (123. + 214. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (134. + 232. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (72.4 - 125. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 501.T + 1.03e5T^{2} \)
53 \( 1 + (-333. - 192. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 548.T + 2.05e5T^{2} \)
61 \( 1 + 218. iT - 2.26e5T^{2} \)
67 \( 1 + 757.T + 3.00e5T^{2} \)
71 \( 1 - 1.10e3iT - 3.57e5T^{2} \)
73 \( 1 + (125. + 72.6i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 891.T + 4.93e5T^{2} \)
83 \( 1 + (745. - 1.29e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-145. - 251. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (144. + 83.6i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−14.55925549675334714646035306898, −12.87021059451656617355350561589, −11.84378458252540474674459908213, −11.01056135829891758864022149064, −10.33888349716884606605603328312, −8.488627216390628161483275175995, −6.83557623682501908805912931764, −5.69297877011703441178625480774, −3.92887027222634333124632740071, −1.45844743488019583643309771577, 1.73704824999091667816507454870, 4.64813446157567125444419698560, 6.12499782920591107973380068350, 7.05494497515574818212466889857, 8.356746671940388627221522838866, 10.29450561296216650134195888375, 11.53964006102858719676360873971, 11.79690924146412460586744137241, 13.51170813318477122278319130558, 14.73302704346380948431887852420

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