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Properties

Label 2-63-63.41-c1-0-3
Degree $2$
Conductor $63$
Sign $0.989 - 0.144i$
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 + (−0.906 + 2.48i)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 + (−0.293 − 1.67i)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 + (−0.342 + 0.939i)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 + (−0.0785 − 0.446i)14-s + (0.100 − 0.983i)15-s + (−0.212 − 0.367i)16-s − 1.44·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.144i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.989 - 0.144i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ 0.989 - 0.144i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.877521 + 0.0638178i\)
\(L(\frac12)\) \(\approx\) \(0.877521 + 0.0638178i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.906 - 2.48i)T \)
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

−15.25289010878114160775682967808, −13.52565603805013017641083402325, −12.97227269790902023890950718830, −12.16412386318947363295813046684, −9.837439584585090778297496859203, −8.950481190633656546737646187534, −8.247860592820650310320152035830, −6.84002042116716227349320229667, −4.79858102474303712610005308165, −2.67006083331478591054067243398, 2.60742785680517495458255626345, 4.57763508591322038403558939724, 6.53249554548544276647403615244, 8.094105212025078339895251011754, 9.422399022012619116663439579587, 10.32488313537306715464052644355, 10.90011066139747276647476212579, 13.32447486167703126218446687718, 13.86248608295219918348242887080, 14.78489842751002289142805644428

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