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Properties

Label 2-63-63.59-c1-0-2
Degree $2$
Conductor $63$
Sign $0.337 + 0.941i$
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  + (−0.254 − 0.146i)2-s + (−1.27 − 1.17i)3-s + (−0.956 − 1.65i)4-s + 3.06·5-s + (0.152 + 0.485i)6-s + (−1.22 − 2.34i)7-s + 1.15i·8-s + (0.254 + 2.98i)9-s + (−0.778 − 0.449i)10-s + 3.89i·11-s + (−0.721 + 3.23i)12-s + (2.02 + 1.17i)13-s + (−0.0325 + 0.776i)14-s + (−3.90 − 3.58i)15-s + (−1.74 + 3.02i)16-s + (1.68 − 2.91i)17-s + ⋯
L(s)  = 1  + (−0.179 − 0.103i)2-s + (−0.736 − 0.676i)3-s + (−0.478 − 0.828i)4-s + 1.36·5-s + (0.0622 + 0.198i)6-s + (−0.463 − 0.886i)7-s + 0.406i·8-s + (0.0848 + 0.996i)9-s + (−0.246 − 0.142i)10-s + 1.17i·11-s + (−0.208 + 0.933i)12-s + (0.562 + 0.324i)13-s + (−0.00869 + 0.207i)14-s + (−1.00 − 0.925i)15-s + (−0.436 + 0.755i)16-s + (0.408 − 0.706i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.337 + 0.941i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ 0.337 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.584600 - 0.411400i\)
\(L(\frac12)\) \(\approx\) \(0.584600 - 0.411400i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.27 + 1.17i)T \)
7 \( 1 + (1.22 + 2.34i)T \)
good2 \( 1 + (0.254 + 0.146i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 3.06T + 5T^{2} \)
11 \( 1 - 3.89iT - 11T^{2} \)
13 \( 1 + (-2.02 - 1.17i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.68 + 2.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.20 + 1.27i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.98iT - 23T^{2} \)
29 \( 1 + (3.67 - 2.12i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.409 - 0.236i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.89 + 6.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.12 - 5.41i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.06 - 3.57i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.02 - 3.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.99 + 2.88i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.34 + 4.05i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.38 + 0.800i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.787 + 1.36i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.6iT - 71T^{2} \)
73 \( 1 + (0.856 + 0.494i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.63 + 8.03i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.49 + 9.51i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.15 - 3.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.98 - 2.87i)T + (48.5 - 84.0i)T^{2} \)
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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

−14.33839151008581993658559941838, −13.60867969042648930958931203017, −12.83855216293988773928513143258, −11.18298640379731392593556607684, −10.08173439636712784399636804440, −9.418698633419927974339541338512, −7.24807532821702596271692829391, −6.10393165918656309466994673722, −4.96097361162037598788362102451, −1.58891858178962163003513947640, 3.39028504049013401082288020391, 5.44622812072426785487193772882, 6.27955897011664389080273747741, 8.532570574340792506934177703297, 9.417433869204116906604185528334, 10.46336616268554097295535781672, 11.91851110323419655842044556031, 12.96211787736945018082440216780, 13.94264097828691121488616305425, 15.46921398023714791506105907564

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