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Properties

Label 2-63-63.5-c3-0-3
Degree $2$
Conductor $63$
Sign $-0.772 + 0.635i$
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  + 5.00i·2-s + (−2.93 + 4.28i)3-s − 17.0·4-s + (9.28 + 16.0i)5-s + (−21.4 − 14.7i)6-s + (12.9 − 13.2i)7-s − 45.5i·8-s + (−9.74 − 25.1i)9-s + (−80.5 + 46.5i)10-s + (30.4 + 17.5i)11-s + (50.2 − 73.2i)12-s + (−3.94 − 2.27i)13-s + (66.2 + 65.0i)14-s + (−96.1 − 7.43i)15-s + 91.5·16-s + (−20.7 − 36.0i)17-s + ⋯
L(s)  = 1  + 1.77i·2-s + (−0.565 + 0.824i)3-s − 2.13·4-s + (0.830 + 1.43i)5-s + (−1.46 − 1.00i)6-s + (0.700 − 0.713i)7-s − 2.01i·8-s + (−0.360 − 0.932i)9-s + (−2.54 + 1.47i)10-s + (0.834 + 0.481i)11-s + (1.20 − 1.76i)12-s + (−0.0841 − 0.0485i)13-s + (1.26 + 1.24i)14-s + (−1.65 − 0.128i)15-s + 1.43·16-s + (−0.296 − 0.513i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 + 0.635i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.772 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.772 + 0.635i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.772 + 0.635i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.410967 - 1.14714i\)
\(L(\frac12)\) \(\approx\) \(0.410967 - 1.14714i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.93 - 4.28i)T \)
7 \( 1 + (-12.9 + 13.2i)T \)
good2 \( 1 - 5.00iT - 8T^{2} \)
5 \( 1 + (-9.28 - 16.0i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-30.4 - 17.5i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (3.94 + 2.27i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (20.7 + 36.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-64.8 - 37.4i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (132. - 76.3i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-42.5 + 24.5i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 101. iT - 2.97e4T^{2} \)
37 \( 1 + (-2.86 + 4.96i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-6.35 + 10.9i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-69.1 - 119. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 538.T + 1.03e5T^{2} \)
53 \( 1 + (105. - 60.8i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 - 312.T + 2.05e5T^{2} \)
61 \( 1 + 504. iT - 2.26e5T^{2} \)
67 \( 1 + 654.T + 3.00e5T^{2} \)
71 \( 1 - 891. iT - 3.57e5T^{2} \)
73 \( 1 + (315. - 182. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 - 485.T + 4.93e5T^{2} \)
83 \( 1 + (-248. - 430. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-685. + 1.18e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-205. + 118. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−15.10141751160803989480930941975, −14.32200604330652046418578327030, −13.83277003673765870069045967877, −11.55612907550307756532411462198, −10.22730687986544135349021144516, −9.388993093243477223092483515969, −7.55916168217390678817277645366, −6.58027065516734226764610215174, −5.57597964666100906173375337987, −4.09391071176062529122951070553, 1.01730752907184294560241202632, 2.08714245116045588124532272321, 4.64036531442158173125239325431, 5.79677771065848477203999649122, 8.453528736481772566056825967524, 9.196885560495015267084463798115, 10.65095159480138511770027655498, 11.96547578403972397836115699946, 12.22094843961547630631383035545, 13.36472992742184677341196937656

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