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Properties

Label 2-63-63.20-c1-0-5
Degree $2$
Conductor $63$
Sign $-0.315 + 0.949i$
Analytic cond. $0.503057$
Root an. cond. $0.709265$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.555 − 0.320i)2-s + (−1.58 − 0.709i)3-s + (−0.794 − 1.37i)4-s + (−1.10 − 1.91i)5-s + (0.650 + 0.901i)6-s + (2.60 − 0.458i)7-s + 2.30i·8-s + (1.99 + 2.24i)9-s + 1.41i·10-s + (−2.93 − 1.69i)11-s + (0.279 + 2.73i)12-s + (1.56 − 0.901i)13-s + (−1.59 − 0.581i)14-s + (0.388 + 3.80i)15-s + (−0.849 + 1.47i)16-s + 5.96·17-s + ⋯
L(s)  = 1  + (−0.392 − 0.226i)2-s + (−0.912 − 0.409i)3-s + (−0.397 − 0.687i)4-s + (−0.494 − 0.856i)5-s + (0.265 + 0.367i)6-s + (0.984 − 0.173i)7-s + 0.813i·8-s + (0.664 + 0.747i)9-s + 0.448i·10-s + (−0.885 − 0.511i)11-s + (0.0806 + 0.790i)12-s + (0.432 − 0.249i)13-s + (−0.426 − 0.155i)14-s + (0.100 + 0.983i)15-s + (−0.212 + 0.367i)16-s + 1.44·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.315 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.315 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.315 + 0.949i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ -0.315 + 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.301886 - 0.418399i\)
\(L(\frac12)\) \(\approx\) \(0.301886 - 0.418399i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.58 + 0.709i)T \)
7 \( 1 + (-2.60 + 0.458i)T \)
good2 \( 1 + (0.555 + 0.320i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (1.10 + 1.91i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.93 + 1.69i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.56 + 0.901i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.96T + 17T^{2} \)
19 \( 1 + 1.64iT - 19T^{2} \)
23 \( 1 + (-2.05 + 1.18i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.44 - 1.41i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (9.28 - 5.36i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.69T + 37T^{2} \)
41 \( 1 + (-0.455 - 0.788i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.96 - 3.39i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.123 - 0.213i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 7.87iT - 53T^{2} \)
59 \( 1 + (-5.39 - 9.33i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.22 + 0.709i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.99 - 6.91i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 - 0.426iT - 73T^{2} \)
79 \( 1 + (-2.49 + 4.31i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.28 + 7.42i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 + (-6.30 - 3.63i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.60492015470906437391607494575, −13.39343325094277448605428053375, −12.27852019808283560307702155894, −11.12062996422734942582304972773, −10.39420090481096450931871626274, −8.685397411203037251136842964272, −7.69995477104068112952787164638, −5.61064527613848862096401696192, −4.81701026673481813818442483916, −1.07411590994453132919694279147, 3.74633195549341848004854403578, 5.30049107908158010805890312216, 7.13823817732467998956881865080, 8.043410538842777296167818821297, 9.683887959524031766078916776385, 10.85908503149562064855125354386, 11.79079881167626059958995202250, 12.85887203843664672889492568508, 14.51372544011439867315837924338, 15.46632794215105622995513474916

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