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Properties

Label 2-63-63.38-c3-0-6
Degree $2$
Conductor $63$
Sign $0.623 - 0.781i$
Analytic cond. $3.71712$
Root an. cond. $1.92798$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.78i·2-s + (2.28 + 4.66i)3-s + 4.80·4-s + (−8.47 + 14.6i)5-s + (8.34 − 4.07i)6-s + (−8.67 + 16.3i)7-s − 22.8i·8-s + (−16.5 + 21.2i)9-s + (26.2 + 15.1i)10-s + (6.18 − 3.57i)11-s + (10.9 + 22.4i)12-s + (67.0 − 38.6i)13-s + (29.2 + 15.5i)14-s + (−87.8 − 6.08i)15-s − 2.49·16-s + (−1.06 + 1.84i)17-s + ⋯
L(s)  = 1  − 0.632i·2-s + (0.438 + 0.898i)3-s + 0.600·4-s + (−0.757 + 1.31i)5-s + (0.567 − 0.277i)6-s + (−0.468 + 0.883i)7-s − 1.01i·8-s + (−0.614 + 0.788i)9-s + (0.829 + 0.478i)10-s + (0.169 − 0.0979i)11-s + (0.263 + 0.539i)12-s + (1.42 − 0.825i)13-s + (0.558 + 0.296i)14-s + (−1.51 − 0.104i)15-s − 0.0390·16-s + (−0.0151 + 0.0263i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.623 - 0.781i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.623 - 0.781i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.47767 + 0.711411i\)
\(L(\frac12)\) \(\approx\) \(1.47767 + 0.711411i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.28 - 4.66i)T \)
7 \( 1 + (8.67 - 16.3i)T \)
good2 \( 1 + 1.78iT - 8T^{2} \)
5 \( 1 + (8.47 - 14.6i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-6.18 + 3.57i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-67.0 + 38.6i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (1.06 - 1.84i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-104. + 60.4i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (9.41 + 5.43i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (110. + 63.9i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 212. iT - 2.97e4T^{2} \)
37 \( 1 + (12.2 + 21.1i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-64.8 - 112. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-86.4 + 149. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 404.T + 1.03e5T^{2} \)
53 \( 1 + (-291. - 168. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + 661.T + 2.05e5T^{2} \)
61 \( 1 - 347. iT - 2.26e5T^{2} \)
67 \( 1 + 269.T + 3.00e5T^{2} \)
71 \( 1 + 637. iT - 3.57e5T^{2} \)
73 \( 1 + (503. + 290. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 171.T + 4.93e5T^{2} \)
83 \( 1 + (-635. + 1.10e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-175. - 303. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-73.3 - 42.3i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−14.99345708385592367633309465306, −13.57529120218853989688390035466, −11.96848705795248446649278101240, −11.06630512180667086707174085836, −10.39928098467373469166622203880, −9.027499058727841957184101844497, −7.48494911774211818271465023164, −5.98567759032345353778405449041, −3.54121157918294016502765560035, −2.86988995690854512563687933663, 1.24412168478698963078654133419, 3.81722709819659291841045737086, 5.92284120534644950299187425930, 7.22616468953642457618387063463, 8.062241767900005351574787253810, 9.176723857318155480231351869524, 11.27269516151948082777565254667, 12.16763807244215627724300226081, 13.30289605391214291503157877916, 14.19838358579456973296963257565

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