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Properties

Label 2-63-63.23-c2-0-11
Degree $2$
Conductor $63$
Sign $-0.788 + 0.614i$
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  − 3.79i·2-s + 3·3-s − 10.3·4-s + (0.282 − 0.163i)5-s − 11.3i·6-s + (−2.88 − 6.37i)7-s + 24.1i·8-s + 9·9-s + (−0.618 − 1.07i)10-s + (13.1 + 7.57i)11-s − 31.0·12-s + (−3.96 + 6.86i)13-s + (−24.1 + 10.9i)14-s + (0.847 − 0.489i)15-s + 49.9·16-s + (6.90 − 3.98i)17-s + ⋯
L(s)  = 1  − 1.89i·2-s + 3-s − 2.59·4-s + (0.0564 − 0.0326i)5-s − 1.89i·6-s + (−0.411 − 0.911i)7-s + 3.01i·8-s + 9-s + (−0.0618 − 0.107i)10-s + (1.19 + 0.688i)11-s − 2.59·12-s + (−0.305 + 0.528i)13-s + (−1.72 + 0.780i)14-s + (0.0564 − 0.0326i)15-s + 3.12·16-s + (0.406 − 0.234i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.614i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.788 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.788 + 0.614i$
Analytic conductor: \(1.71662\)
Root analytic conductor: \(1.31020\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1),\ -0.788 + 0.614i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.449554 - 1.30773i\)
\(L(\frac12)\) \(\approx\) \(0.449554 - 1.30773i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
7 \( 1 + (2.88 + 6.37i)T \)
good2 \( 1 + 3.79iT - 4T^{2} \)
5 \( 1 + (-0.282 + 0.163i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-13.1 - 7.57i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (3.96 - 6.86i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-6.90 + 3.98i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-2.86 + 4.96i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-1.79 + 1.03i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (5.71 - 3.30i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + 50.5T + 961T^{2} \)
37 \( 1 + (-5.56 + 9.63i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (12.8 + 7.43i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-3.67 - 6.37i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 - 66.9iT - 2.20e3T^{2} \)
53 \( 1 + (-22.0 + 12.7i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 - 39.7iT - 3.48e3T^{2} \)
61 \( 1 + 74.3T + 3.72e3T^{2} \)
67 \( 1 - 42.9T + 4.48e3T^{2} \)
71 \( 1 + 72.5iT - 5.04e3T^{2} \)
73 \( 1 + (45.8 + 79.3i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 - 23.0T + 6.24e3T^{2} \)
83 \( 1 + (-97.9 + 56.5i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (63.4 + 36.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (57.2 + 99.0i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.93763573273747816293522316639, −13.04926671491040131158985155789, −12.09599844968144621120814836631, −10.82178108831080405006389912340, −9.593560934099516524790247385139, −9.198263143174450215469377305383, −7.38280039159151916255266785543, −4.39909286002029260629509316660, −3.41137255078075799544315104222, −1.63350080380024774622735167245, 3.74049956416502429264579993066, 5.58153286454495086032529823473, 6.75793673744395542556662682824, 8.059843623299195237151889396124, 8.915372835981668623127229108784, 9.770971539234730880551863713584, 12.39039971920360435587278054910, 13.51031264305693280212481029883, 14.46500150399618244403339923888, 15.06456060615678084698557513420

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