Properties

Label 2-63-63.4-c1-0-0
Degree $2$
Conductor $63$
Sign $-0.296 - 0.954i$
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  + (−0.5 + 0.866i)2-s + (−1.5 + 0.866i)3-s + (0.500 + 0.866i)4-s − 5-s − 1.73i·6-s + (0.5 + 2.59i)7-s − 3·8-s + (1.5 − 2.59i)9-s + (0.5 − 0.866i)10-s + 5·11-s + (−1.5 − 0.866i)12-s + (2.5 − 4.33i)13-s + (−2.5 − 0.866i)14-s + (1.5 − 0.866i)15-s + (0.500 − 0.866i)16-s + (−1.5 + 2.59i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.866 + 0.499i)3-s + (0.250 + 0.433i)4-s − 0.447·5-s − 0.707i·6-s + (0.188 + 0.981i)7-s − 1.06·8-s + (0.5 − 0.866i)9-s + (0.158 − 0.273i)10-s + 1.50·11-s + (−0.433 − 0.249i)12-s + (0.693 − 1.20i)13-s + (−0.668 − 0.231i)14-s + (0.387 − 0.223i)15-s + (0.125 − 0.216i)16-s + (−0.363 + 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.296 - 0.954i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ -0.296 - 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.368746 + 0.500724i\)
\(L(\frac12)\) \(\approx\) \(0.368746 + 0.500724i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 2.59i)T \)
good2 \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + T + 5T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3T + 23T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.5 + 4.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7 + 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (1.5 - 2.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.5 - 11.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.5 - 7.79i)T + (-48.5 + 84.0i)T^{2} \)
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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

−15.47674288230236081681088806183, −14.87376368474501683723996129600, −12.70978823332168147455043032808, −11.86846019163676428235716632122, −11.02430094162689954314665678942, −9.293965850403350358128547671807, −8.316946475014102569650881926035, −6.71366573090050386876404957274, −5.69288996773464311684903806541, −3.72236946875674149886478688528, 1.34481939087700784078345092839, 4.24046345121247578862061925991, 6.25357711914196792490783242610, 7.16140752125138540149038860546, 9.058662928250434748634489185049, 10.39171380354547441234432551552, 11.56480729303810202545692114285, 11.71708196006409488967224957758, 13.46385107895923437615042734149, 14.51799207339761565762672581044

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