Properties

Label 2-63-63.4-c3-0-14
Degree $2$
Conductor $63$
Sign $0.734 + 0.678i$
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  + (1.46 − 2.54i)2-s + (4.92 + 1.65i)3-s + (−0.305 − 0.529i)4-s − 3.69·5-s + (11.4 − 10.0i)6-s + (12.3 − 13.8i)7-s + 21.6·8-s + (21.4 + 16.3i)9-s + (−5.42 + 9.39i)10-s − 65.7·11-s + (−0.626 − 3.11i)12-s + (2.73 − 4.74i)13-s + (−16.9 − 51.6i)14-s + (−18.2 − 6.13i)15-s + (34.2 − 59.3i)16-s + (−25.1 + 43.5i)17-s + ⋯
L(s)  = 1  + (0.518 − 0.898i)2-s + (0.947 + 0.319i)3-s + (−0.0382 − 0.0662i)4-s − 0.330·5-s + (0.778 − 0.685i)6-s + (0.666 − 0.745i)7-s + 0.958·8-s + (0.795 + 0.605i)9-s + (−0.171 + 0.297i)10-s − 1.80·11-s + (−0.0150 − 0.0749i)12-s + (0.0584 − 0.101i)13-s + (−0.323 − 0.985i)14-s + (−0.313 − 0.105i)15-s + (0.535 − 0.927i)16-s + (−0.358 + 0.621i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.734 + 0.678i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.734 + 0.678i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.26767 - 0.886718i\)
\(L(\frac12)\) \(\approx\) \(2.26767 - 0.886718i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-12.3 + 13.8i)T \)
good2 \( 1 + (-1.46 + 2.54i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + 3.69T + 125T^{2} \)
11 \( 1 + 65.7T + 1.33e3T^{2} \)
13 \( 1 + (-2.73 + 4.74i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (25.1 - 43.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-0.769 - 1.33i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 120.T + 1.21e4T^{2} \)
29 \( 1 + (39.2 + 68.0i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-151. - 262. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-96.6 - 167. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-196. + 340. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (138. + 239. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-126. + 218. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (204. - 353. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (131. + 227. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-56.1 + 97.2i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-49.1 - 85.1i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 255.T + 3.57e5T^{2} \)
73 \( 1 + (-344. + 596. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-542. + 939. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-152. - 263. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-550. - 953. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-493. - 855. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.90354170690395565349774179989, −13.38303284149352343028680101565, −12.19961647811642387626897047344, −10.74848983604842003673236597846, −10.24151144112092458628701134641, −8.204196922902559581943761617982, −7.58069912438241956970298278359, −4.81169388978896630496223351169, −3.66149240047020816178098867069, −2.17044391967789407382330913712, 2.35016298971812639031798808618, 4.56296557571688192515591452369, 5.95342810291360295633967784332, 7.63920251459518517537098297153, 8.106976314787953499433413923169, 9.801527296997346846838469640388, 11.32181514949756554525858695396, 12.85184272234082054511078100347, 13.72390897789620005588522746552, 14.71420982520893917283485327089

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