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Properties

Label 2-63-63.23-c2-0-13
Degree $2$
Conductor $63$
Sign $-0.894 - 0.446i$
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  − 2.87i·2-s + (−2.93 − 0.599i)3-s − 4.24·4-s + (−6.53 + 3.77i)5-s + (−1.72 + 8.44i)6-s + (2.05 − 6.69i)7-s + 0.709i·8-s + (8.28 + 3.52i)9-s + (10.8 + 18.7i)10-s + (−13.1 − 7.59i)11-s + (12.4 + 2.54i)12-s + (4.30 − 7.45i)13-s + (−19.2 − 5.90i)14-s + (21.4 − 7.17i)15-s − 14.9·16-s + (−4.60 + 2.66i)17-s + ⋯
L(s)  = 1  − 1.43i·2-s + (−0.979 − 0.199i)3-s − 1.06·4-s + (−1.30 + 0.754i)5-s + (−0.286 + 1.40i)6-s + (0.293 − 0.955i)7-s + 0.0886i·8-s + (0.920 + 0.391i)9-s + (1.08 + 1.87i)10-s + (−1.19 − 0.690i)11-s + (1.04 + 0.212i)12-s + (0.331 − 0.573i)13-s + (−1.37 − 0.421i)14-s + (1.43 − 0.478i)15-s − 0.934·16-s + (−0.271 + 0.156i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.446i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.894 - 0.446i$
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: \((2,\ 63,\ (\ :1),\ -0.894 - 0.446i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.114141 + 0.484698i\)
\(L(\frac12)\) \(\approx\) \(0.114141 + 0.484698i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.93 + 0.599i)T \)
7 \( 1 + (-2.05 + 6.69i)T \)
good2 \( 1 + 2.87iT - 4T^{2} \)
5 \( 1 + (6.53 - 3.77i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (13.1 + 7.59i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-4.30 + 7.45i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (4.60 - 2.66i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (0.417 - 0.722i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-33.8 + 19.5i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-12.5 + 7.25i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 12.7T + 961T^{2} \)
37 \( 1 + (11.7 - 20.3i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (13.4 + 7.75i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-0.448 - 0.776i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + 2.35iT - 2.20e3T^{2} \)
53 \( 1 + (31.4 - 18.1i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + 48.6iT - 3.48e3T^{2} \)
61 \( 1 + 29.6T + 3.72e3T^{2} \)
67 \( 1 - 92.5T + 4.48e3T^{2} \)
71 \( 1 - 15.1iT - 5.04e3T^{2} \)
73 \( 1 + (46.8 + 81.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + 82.1T + 6.24e3T^{2} \)
83 \( 1 + (-127. + 73.3i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-92.3 - 53.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (26.0 + 45.0i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.54404591655057303220448707059, −12.64775426213615962049255522311, −11.48973538947100087706282722076, −10.81727664390609053434988370964, −10.41371990197610202879685720993, −8.049431440468640334504178840223, −6.83515382450712050301286010330, −4.63549392031665671306821864464, −3.22424197911385682344932574576, −0.52148987682429679346737572832, 4.64817414435236099893798663826, 5.37419393783963400787342576645, 6.94971681062098608054258948714, 8.010580484355659562348967519937, 9.129382956516028007732193436454, 11.10915189497396338730719081552, 12.01594526519611641049095567181, 13.08049999204700168644113594754, 14.96606081085873893862400389288, 15.69542239603633568335693625305

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