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Properties

Label 2-63-9.7-c3-0-0
Degree $2$
Conductor $63$
Sign $-0.795 - 0.605i$
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  + (−1.21 + 2.11i)2-s + (−2.70 − 4.43i)3-s + (1.03 + 1.78i)4-s + (2.61 + 4.53i)5-s + (12.6 − 0.303i)6-s + (−3.5 + 6.06i)7-s − 24.5·8-s + (−12.3 + 24.0i)9-s − 12.7·10-s + (−30.6 + 53.1i)11-s + (5.13 − 9.40i)12-s + (0.911 + 1.57i)13-s + (−8.52 − 14.7i)14-s + (13.0 − 23.8i)15-s + (21.6 − 37.4i)16-s − 76.2·17-s + ⋯
L(s)  = 1  + (−0.430 + 0.746i)2-s + (−0.520 − 0.853i)3-s + (0.128 + 0.223i)4-s + (0.234 + 0.405i)5-s + (0.861 − 0.0206i)6-s + (−0.188 + 0.327i)7-s − 1.08·8-s + (−0.457 + 0.888i)9-s − 0.403·10-s + (−0.840 + 1.45i)11-s + (0.123 − 0.226i)12-s + (0.0194 + 0.0336i)13-s + (−0.162 − 0.281i)14-s + (0.224 − 0.411i)15-s + (0.337 − 0.585i)16-s − 1.08·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.795 - 0.605i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.795 - 0.605i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.795 - 0.605i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.220761 + 0.654922i\)
\(L(\frac12)\) \(\approx\) \(0.220761 + 0.654922i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.70 + 4.43i)T \)
7 \( 1 + (3.5 - 6.06i)T \)
good2 \( 1 + (1.21 - 2.11i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (-2.61 - 4.53i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (30.6 - 53.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-0.911 - 1.57i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 76.2T + 4.91e3T^{2} \)
19 \( 1 - 75.2T + 6.85e3T^{2} \)
23 \( 1 + (-65.7 - 113. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-32.0 + 55.4i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-88.7 - 153. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 337.T + 5.06e4T^{2} \)
41 \( 1 + (22.8 + 39.5i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-146. + 254. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-163. + 282. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 66.4T + 1.48e5T^{2} \)
59 \( 1 + (-344. - 597. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (336. - 582. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-288. - 499. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 4.08T + 3.57e5T^{2} \)
73 \( 1 - 747.T + 3.89e5T^{2} \)
79 \( 1 + (531. - 920. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-415. + 720. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 460.T + 7.04e5T^{2} \)
97 \( 1 + (431. - 747. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−15.26282039593881115666871840936, −13.73705312256271341567784046042, −12.59009107531596588437110503700, −11.76498881168803843610942072164, −10.29957427463294377891814073475, −8.755618283582829140146577215140, −7.37862678323798732072171500360, −6.79341916768049689486004297810, −5.35264784998387323571299780227, −2.47962668650558750285892523886, 0.55701310006217309717117682467, 3.11939027077760415601570324107, 5.11242726171184086046054705222, 6.32717276383023565858939754148, 8.646422775908464977555573011099, 9.604207221208478803296893002448, 10.78520812038813161682455330539, 11.21632825275730775956820199755, 12.65453091998179317582109342623, 13.99062741592787044436204021816

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