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Properties

Label 2-63-63.32-c2-0-5
Degree $2$
Conductor $63$
Sign $0.694 - 0.719i$
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  + (1.11 + 0.642i)2-s + (2.34 + 1.86i)3-s + (−1.17 − 2.03i)4-s + 7.87i·5-s + (1.41 + 3.58i)6-s + (0.417 − 6.98i)7-s − 8.15i·8-s + (2.03 + 8.76i)9-s + (−5.06 + 8.76i)10-s − 12.3i·11-s + (1.03 − 6.96i)12-s + (−3.39 + 5.87i)13-s + (4.95 − 7.50i)14-s + (−14.6 + 18.5i)15-s + (0.548 − 0.949i)16-s + (6.19 + 3.57i)17-s + ⋯
L(s)  = 1  + (0.556 + 0.321i)2-s + (0.783 + 0.621i)3-s + (−0.293 − 0.508i)4-s + 1.57i·5-s + (0.236 + 0.597i)6-s + (0.0596 − 0.998i)7-s − 1.01i·8-s + (0.226 + 0.974i)9-s + (−0.506 + 0.876i)10-s − 1.12i·11-s + (0.0862 − 0.580i)12-s + (−0.261 + 0.452i)13-s + (0.353 − 0.536i)14-s + (−0.979 + 1.23i)15-s + (0.0342 − 0.0593i)16-s + (0.364 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.694 - 0.719i$
Analytic conductor: \(1.71662\)
Root analytic conductor: \(1.31020\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1),\ 0.694 - 0.719i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.61088 + 0.684466i\)
\(L(\frac12)\) \(\approx\) \(1.61088 + 0.684466i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.34 - 1.86i)T \)
7 \( 1 + (-0.417 + 6.98i)T \)
good2 \( 1 + (-1.11 - 0.642i)T + (2 + 3.46i)T^{2} \)
5 \( 1 - 7.87iT - 25T^{2} \)
11 \( 1 + 12.3iT - 121T^{2} \)
13 \( 1 + (3.39 - 5.87i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-6.19 - 3.57i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (15.4 + 26.8i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 - 2.16iT - 529T^{2} \)
29 \( 1 + (8.97 - 5.17i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-10.0 - 17.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-2.46 - 4.26i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-28.0 - 16.1i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-25.5 - 44.1i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-18.5 - 10.7i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-51.0 - 29.4i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (89.6 - 51.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (4.46 - 7.74i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (45.3 + 78.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 2.12iT - 5.04e3T^{2} \)
73 \( 1 + (-35.2 + 61.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-72.3 + 125. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (13.6 - 7.85i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-4.08 + 2.35i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (47.0 + 81.4i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−14.67334384428117688047836679083, −14.01008024990347725690679754779, −13.37884870407868643664190259641, −10.95675996687122808656202872858, −10.48266228613170799006531714575, −9.210108221735490576017003424538, −7.46689905530641249541725130050, −6.31447613460387230382708402325, −4.43362541168024964743387703072, −3.15505846622446309321517882608, 2.15807759223748894200042333562, 4.13136868747966340001016022547, 5.51180170546287264283975212562, 7.80477508610360020536499802944, 8.594549646496345546707962317161, 9.557807349618490460019997829645, 12.13577859606039074581581553399, 12.39241648811926505079146418740, 13.10968890060266136622835944849, 14.37027856577082783530227173837

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