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Properties

Label 2-63-63.20-c3-0-13
Degree $2$
Conductor $63$
Sign $-0.108 + 0.994i$
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  + (0.628 + 0.362i)2-s + (−2.99 + 4.24i)3-s + (−3.73 − 6.47i)4-s + (−5.53 − 9.58i)5-s + (−3.42 + 1.58i)6-s + (13.3 − 12.8i)7-s − 11.2i·8-s + (−9.10 − 25.4i)9-s − 8.03i·10-s + (0.219 + 0.126i)11-s + (38.6 + 3.48i)12-s + (−12.3 + 7.15i)13-s + (13.0 − 3.26i)14-s + (57.2 + 5.15i)15-s + (−25.8 + 44.7i)16-s − 92.5·17-s + ⋯
L(s)  = 1  + (0.222 + 0.128i)2-s + (−0.575 + 0.817i)3-s + (−0.467 − 0.808i)4-s + (−0.494 − 0.857i)5-s + (−0.232 + 0.107i)6-s + (0.718 − 0.695i)7-s − 0.496i·8-s + (−0.337 − 0.941i)9-s − 0.253i·10-s + (0.00602 + 0.00347i)11-s + (0.930 + 0.0838i)12-s + (−0.264 + 0.152i)13-s + (0.248 − 0.0622i)14-s + (0.985 + 0.0888i)15-s + (−0.403 + 0.698i)16-s − 1.32·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.108 + 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.108 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.108 + 0.994i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.108 + 0.994i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.571628 - 0.637621i\)
\(L(\frac12)\) \(\approx\) \(0.571628 - 0.637621i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.99 - 4.24i)T \)
7 \( 1 + (-13.3 + 12.8i)T \)
good2 \( 1 + (-0.628 - 0.362i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (5.53 + 9.58i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-0.219 - 0.126i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (12.3 - 7.15i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 92.5T + 4.91e3T^{2} \)
19 \( 1 + 130. iT - 6.85e3T^{2} \)
23 \( 1 + (102. - 59.3i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-248. - 143. i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-214. + 123. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 188.T + 5.06e4T^{2} \)
41 \( 1 + (-53.2 - 92.2i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (21.5 - 37.3i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (68.8 - 119. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 419. iT - 1.48e5T^{2} \)
59 \( 1 + (217. + 376. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (163. + 94.3i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (185. + 321. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 26.1iT - 3.57e5T^{2} \)
73 \( 1 - 728. iT - 3.89e5T^{2} \)
79 \( 1 + (48.6 - 84.3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-401. + 695. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 236.T + 7.04e5T^{2} \)
97 \( 1 + (-1.26e3 - 732. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−14.27734359000304234348589926847, −13.19638471097445331260278833990, −11.71158130722884320415326234126, −10.77923490901380440474515665066, −9.599754432407758726274725071124, −8.493766051597122905495033360621, −6.52503197550332287638316442977, −4.78611321938994744768295621753, −4.46707313524555258545941163498, −0.60389137709429538346149519275, 2.53873494886484665176150758521, 4.53766116399526029551649261324, 6.24174294241584239964760718714, 7.68418421331663797051986448931, 8.467192268500192449491771811721, 10.58126984988852280866923524265, 11.81835626729349954581797106150, 12.20901407161870533117400495073, 13.59201069582938645703440218884, 14.50771773438695815534921053021

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