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Properties

Label 2-63-63.41-c3-0-5
Degree $2$
Conductor $63$
Sign $-0.949 - 0.313i$
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  + (−3.38 + 1.95i)2-s + (4.35 + 2.83i)3-s + (3.63 − 6.29i)4-s + (−5.82 + 10.0i)5-s + (−20.2 − 1.07i)6-s + (17.1 + 7.01i)7-s − 2.86i·8-s + (10.9 + 24.6i)9-s − 45.5i·10-s + (−41.2 + 23.8i)11-s + (33.6 − 17.1i)12-s + (−60.2 − 34.7i)13-s + (−71.7 + 9.76i)14-s + (−53.9 + 27.4i)15-s + (34.6 + 60.0i)16-s − 25.6·17-s + ⋯
L(s)  = 1  + (−1.19 + 0.690i)2-s + (0.838 + 0.545i)3-s + (0.454 − 0.786i)4-s + (−0.521 + 0.902i)5-s + (−1.37 − 0.0729i)6-s + (0.925 + 0.378i)7-s − 0.126i·8-s + (0.405 + 0.913i)9-s − 1.43i·10-s + (−1.13 + 0.653i)11-s + (0.809 − 0.412i)12-s + (−1.28 − 0.741i)13-s + (−1.36 + 0.186i)14-s + (−0.928 + 0.472i)15-s + (0.541 + 0.938i)16-s − 0.366·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.313i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.949 - 0.313i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.949 - 0.313i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.129382 + 0.803324i\)
\(L(\frac12)\) \(\approx\) \(0.129382 + 0.803324i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.35 - 2.83i)T \)
7 \( 1 + (-17.1 - 7.01i)T \)
good2 \( 1 + (3.38 - 1.95i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (5.82 - 10.0i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (41.2 - 23.8i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (60.2 + 34.7i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 25.6T + 4.91e3T^{2} \)
19 \( 1 + 41.1iT - 6.85e3T^{2} \)
23 \( 1 + (-145. - 84.0i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-28.2 + 16.2i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-148. - 85.7i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 3.86T + 5.06e4T^{2} \)
41 \( 1 + (-168. + 292. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-207. - 359. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-74.3 - 128. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 59.7iT - 1.48e5T^{2} \)
59 \( 1 + (-282. + 488. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (571. - 330. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-145. + 252. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 3.05iT - 3.57e5T^{2} \)
73 \( 1 - 506. iT - 3.89e5T^{2} \)
79 \( 1 + (96.6 + 167. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (216. + 374. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 924.T + 7.04e5T^{2} \)
97 \( 1 + (-943. + 544. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−15.38447859539421750954486348953, −14.48477913448752018778006047627, −12.84015105733013843970588791687, −10.99787868292288915259090922663, −10.14176783165612983672479480264, −8.990880472817980436948566121527, −7.74312788508992622696656773276, −7.35035203926356511682776996653, −4.90764646427383889637358556742, −2.72659732255093250090610493959, 0.76088170028885496830022278940, 2.42490853736312737225629181409, 4.73924925744347925633135027250, 7.46493843180089067949873037124, 8.271857261545212956125904642019, 9.034697001978792999841661290164, 10.37778575948427778427659891812, 11.62362181502212359706094148474, 12.60048089021954529357933028787, 13.90325698013147067698556779172

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