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Properties

Label 2-63-63.47-c1-0-4
Degree $2$
Conductor $63$
Sign $-0.916 + 0.400i$
Analytic cond. $0.503057$
Root an. cond. $0.709265$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)2-s + (−1.5 − 0.866i)3-s + (0.5 − 0.866i)4-s − 3·5-s + 3·6-s + (−2.5 − 0.866i)7-s − 1.73i·8-s + (1.5 + 2.59i)9-s + (4.5 − 2.59i)10-s + 1.73i·11-s + (−1.5 + 0.866i)12-s + (−1.5 + 0.866i)13-s + (4.5 − 0.866i)14-s + (4.5 + 2.59i)15-s + (2.49 + 4.33i)16-s + (1.5 + 2.59i)17-s + ⋯
L(s)  = 1  + (−1.06 + 0.612i)2-s + (−0.866 − 0.499i)3-s + (0.250 − 0.433i)4-s − 1.34·5-s + 1.22·6-s + (−0.944 − 0.327i)7-s − 0.612i·8-s + (0.5 + 0.866i)9-s + (1.42 − 0.821i)10-s + 0.522i·11-s + (−0.433 + 0.250i)12-s + (−0.416 + 0.240i)13-s + (1.20 − 0.231i)14-s + (1.16 + 0.670i)15-s + (0.624 + 1.08i)16-s + (0.363 + 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.916 + 0.400i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ -0.916 + 0.400i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 + 0.866i)T \)
good2 \( 1 + (1.5 - 0.866i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 3T + 5T^{2} \)
11 \( 1 - 1.73iT - 11T^{2} \)
13 \( 1 + (1.5 - 0.866i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.5 + 2.59i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.19iT - 23T^{2} \)
29 \( 1 + (4.5 + 2.59i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3 + 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.5 - 4.33i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-12 + 6.92i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + (4.5 - 2.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.5 - 12.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.5 + 0.866i)T + (48.5 + 84.0i)T^{2} \)
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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

−15.06020113073052208804577183466, −12.97195490744740051471590904968, −12.33391187919379767374474263630, −10.96934280226174371830932762481, −9.835219782869175814349722782671, −8.288463075388603889105459389983, −7.28822556449389205247566826803, −6.47914789526070544924075624035, −4.19208744532054610463242934762, 0, 3.59247465438609684224541002929, 5.52642751154064048615476871412, 7.34549107508382060348600072128, 8.802172192593933841385547815591, 9.864422386167622675789850927708, 10.92088681910152773336161361167, 11.74329413950311306917841786480, 12.63168405721202051569981545816, 14.72518720460067095407821569932

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