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Properties

Label 2-63-63.31-c2-0-1
Degree $2$
Conductor $63$
Sign $0.279 - 0.960i$
Analytic cond. $1.71662$
Root an. cond. $1.31020$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.227 + 0.394i)2-s + (−2.69 − 1.30i)3-s + (1.89 + 3.28i)4-s + 4.37i·5-s + (1.13 − 0.766i)6-s + (5.22 + 4.66i)7-s − 3.54·8-s + (5.57 + 7.06i)9-s + (−1.72 − 0.994i)10-s − 0.139·11-s + (−0.819 − 11.3i)12-s + (−1.71 − 0.987i)13-s + (−3.02 + 0.996i)14-s + (5.72 − 11.7i)15-s + (−6.77 + 11.7i)16-s + (−26.7 − 15.4i)17-s + ⋯
L(s)  = 1  + (−0.113 + 0.197i)2-s + (−0.899 − 0.436i)3-s + (0.474 + 0.821i)4-s + 0.874i·5-s + (0.188 − 0.127i)6-s + (0.745 + 0.666i)7-s − 0.443·8-s + (0.619 + 0.785i)9-s + (−0.172 − 0.0994i)10-s − 0.0126·11-s + (−0.0683 − 0.945i)12-s + (−0.131 − 0.0759i)13-s + (−0.216 + 0.0711i)14-s + (0.381 − 0.786i)15-s + (−0.423 + 0.733i)16-s + (−1.57 − 0.909i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.279 - 0.960i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.279 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.279 - 0.960i$
Analytic conductor: \(1.71662\)
Root analytic conductor: \(1.31020\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1),\ 0.279 - 0.960i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.780005 + 0.585144i\)
\(L(\frac12)\) \(\approx\) \(0.780005 + 0.585144i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.69 + 1.30i)T \)
7 \( 1 + (-5.22 - 4.66i)T \)
good2 \( 1 + (0.227 - 0.394i)T + (-2 - 3.46i)T^{2} \)
5 \( 1 - 4.37iT - 25T^{2} \)
11 \( 1 + 0.139T + 121T^{2} \)
13 \( 1 + (1.71 + 0.987i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (26.7 + 15.4i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-25.2 + 14.5i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 - 29.7T + 529T^{2} \)
29 \( 1 + (-7.28 - 12.6i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (6.82 - 3.94i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (7.73 + 13.3i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-0.747 - 0.431i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (15.6 + 27.1i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-58.0 - 33.4i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-16.9 + 29.3i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-57.4 + 33.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (35.9 + 20.7i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (51.7 + 89.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 86.4T + 5.04e3T^{2} \)
73 \( 1 + (28.6 + 16.5i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (24.3 - 42.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (102. - 59.4i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (33.1 - 19.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-70.3 + 40.6i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−15.25327777972161811298757579476, −13.73683219910901594411155851118, −12.50080686695973950220311911607, −11.40576683241105103367060315179, −10.99699942212241452681758323888, −8.951433118344297101039784671225, −7.39223584461253305173428087103, −6.73052010605544557945206546422, −5.06383083171879071002072959217, −2.61957133700842662158381184626, 1.18183630518812640405228522140, 4.48118955746234351847596143558, 5.56168483101769852722907418885, 7.01359058065097700092259900178, 8.899025490177473410705230203711, 10.17346776170876205233491913977, 11.05404836568123462270829318187, 11.92095090628488893442976487367, 13.27876392844933561470213183255, 14.73071712215195852213788569244

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