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Properties

Label 2-63-63.41-c3-0-11
Degree $2$
Conductor $63$
Sign $0.986 + 0.165i$
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  + (−2.31 + 1.33i)2-s + (3.10 − 4.16i)3-s + (−0.417 + 0.723i)4-s + (0.223 − 0.386i)5-s + (−1.62 + 13.8i)6-s + (18.4 + 1.96i)7-s − 23.6i·8-s + (−7.70 − 25.8i)9-s + 1.19i·10-s + (34.2 − 19.7i)11-s + (1.71 + 3.98i)12-s + (68.4 + 39.5i)13-s + (−45.3 + 20.0i)14-s + (−0.917 − 2.13i)15-s + (28.3 + 49.0i)16-s + 9.74·17-s + ⋯
L(s)  = 1  + (−0.819 + 0.473i)2-s + (0.597 − 0.801i)3-s + (−0.0522 + 0.0904i)4-s + (0.0199 − 0.0345i)5-s + (−0.110 + 0.939i)6-s + (0.994 + 0.106i)7-s − 1.04i·8-s + (−0.285 − 0.958i)9-s + 0.0377i·10-s + (0.939 − 0.542i)11-s + (0.0412 + 0.0959i)12-s + (1.46 + 0.843i)13-s + (−0.865 + 0.383i)14-s + (−0.0157 − 0.0366i)15-s + (0.442 + 0.766i)16-s + 0.139·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.165i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.986 + 0.165i$
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: \((2,\ 63,\ (\ :3/2),\ 0.986 + 0.165i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.21582 - 0.101132i\)
\(L(\frac12)\) \(\approx\) \(1.21582 - 0.101132i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.10 + 4.16i)T \)
7 \( 1 + (-18.4 - 1.96i)T \)
good2 \( 1 + (2.31 - 1.33i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (-0.223 + 0.386i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-34.2 + 19.7i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-68.4 - 39.5i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 9.74T + 4.91e3T^{2} \)
19 \( 1 + 73.1iT - 6.85e3T^{2} \)
23 \( 1 + (126. + 73.0i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (134. - 77.4i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-9.87 - 5.70i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 337.T + 5.06e4T^{2} \)
41 \( 1 + (-53.3 + 92.3i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (45.6 + 78.9i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-276. - 479. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 239. iT - 1.48e5T^{2} \)
59 \( 1 + (-126. + 218. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (342. - 197. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-13.5 + 23.4i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 348. iT - 3.57e5T^{2} \)
73 \( 1 + 923. iT - 3.89e5T^{2} \)
79 \( 1 + (-280. - 485. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (281. + 487. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 644.T + 7.04e5T^{2} \)
97 \( 1 + (-427. + 246. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−14.30651375286039138367489741380, −13.55251815891855909981361561609, −12.22298642611675756970761510042, −11.07543931251262074453317501786, −9.014881283948808555357073996670, −8.694179655456849333599492014771, −7.44813945678373977191651209334, −6.31706048477695842784856772066, −3.82090894739272451093581813637, −1.34260282923781712715731697488, 1.70391956353763982591151372233, 3.93774695779105160420667222702, 5.55417301179473895212469854610, 7.995912265251684794744322333755, 8.727008087107431862899534759327, 9.949149632733679191726795001050, 10.73575936476074481517539655734, 11.79090676493985933015179438181, 13.76006857139705375405617880874, 14.49715264548579811070361390523

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