Properties

Label 2-63-63.16-c3-0-3
Degree $2$
Conductor $63$
Sign $-0.821 - 0.569i$
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  + (1.68 + 2.92i)2-s + (0.0520 + 5.19i)3-s + (−1.68 + 2.92i)4-s − 9.74·5-s + (−15.0 + 8.91i)6-s + (0.158 + 18.5i)7-s + 15.5·8-s + (−26.9 + 0.540i)9-s + (−16.4 − 28.4i)10-s + 38.9·11-s + (−15.2 − 8.62i)12-s + (−31.2 − 54.1i)13-s + (−53.8 + 31.6i)14-s + (−0.506 − 50.6i)15-s + (39.8 + 68.9i)16-s + (63.3 + 109. i)17-s + ⋯
L(s)  = 1  + (0.596 + 1.03i)2-s + (0.0100 + 0.999i)3-s + (−0.211 + 0.365i)4-s − 0.871·5-s + (−1.02 + 0.606i)6-s + (0.00854 + 0.999i)7-s + 0.688·8-s + (−0.999 + 0.0200i)9-s + (−0.519 − 0.900i)10-s + 1.06·11-s + (−0.367 − 0.207i)12-s + (−0.667 − 1.15i)13-s + (−1.02 + 0.605i)14-s + (−0.00872 − 0.871i)15-s + (0.621 + 1.07i)16-s + (0.903 + 1.56i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.569i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.821 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.821 - 0.569i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.821 - 0.569i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.525602 + 1.68068i\)
\(L(\frac12)\) \(\approx\) \(0.525602 + 1.68068i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0520 - 5.19i)T \)
7 \( 1 + (-0.158 - 18.5i)T \)
good2 \( 1 + (-1.68 - 2.92i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + 9.74T + 125T^{2} \)
11 \( 1 - 38.9T + 1.33e3T^{2} \)
13 \( 1 + (31.2 + 54.1i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-63.3 - 109. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-22.7 + 39.3i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 - 153.T + 1.21e4T^{2} \)
29 \( 1 + (-27.4 + 47.5i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (54.8 - 95.0i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-144. + 250. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (11.3 + 19.5i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-23.9 + 41.4i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (193. + 334. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (133. + 231. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (193. - 334. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (17.8 + 30.9i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (17.8 - 30.9i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 146.T + 3.57e5T^{2} \)
73 \( 1 + (364. + 631. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (250. + 434. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-169. + 293. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (104. - 180. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (56.7 - 98.2i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−14.97647105552173473123957282342, −14.56750038861897033718558699800, −12.77507348381419340430344166604, −11.60177154234786079499538688051, −10.34902658748748740302774484712, −8.860959440728618510304670261459, −7.70833562916040429566841312940, −6.03394759341665845440233281236, −4.99256263903097054788396398003, −3.56942772398857387904798229516, 1.18342174549559027604568142639, 3.19188394048030449135907244278, 4.57234470101286396236186776592, 6.96782098261591154100444443707, 7.67846663773034062917819135583, 9.567157677519905125266823050220, 11.38297233313981963766412205364, 11.66932310866461222677423942542, 12.68951731193394662053975368838, 13.88780277826751139756201959043

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