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Properties

Label 2-63-63.38-c3-0-8
Degree $2$
Conductor $63$
Sign $0.337 - 0.941i$
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  + 1.10i·2-s + (0.171 + 5.19i)3-s + 6.77·4-s + (6.23 − 10.8i)5-s + (−5.74 + 0.189i)6-s + (11.7 + 14.3i)7-s + 16.3i·8-s + (−26.9 + 1.77i)9-s + (11.9 + 6.90i)10-s + (3.22 − 1.86i)11-s + (1.15 + 35.1i)12-s + (−68.0 + 39.2i)13-s + (−15.8 + 12.9i)14-s + (57.1 + 30.5i)15-s + 36.0·16-s + (56.8 − 98.4i)17-s + ⋯
L(s)  = 1  + 0.391i·2-s + (0.0329 + 0.999i)3-s + 0.846·4-s + (0.557 − 0.966i)5-s + (−0.391 + 0.0128i)6-s + (0.632 + 0.774i)7-s + 0.722i·8-s + (−0.997 + 0.0658i)9-s + (0.378 + 0.218i)10-s + (0.0883 − 0.0510i)11-s + (0.0278 + 0.846i)12-s + (−1.45 + 0.838i)13-s + (−0.303 + 0.247i)14-s + (0.984 + 0.525i)15-s + 0.563·16-s + (0.810 − 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.337 - 0.941i$
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: \((2,\ 63,\ (\ :3/2),\ 0.337 - 0.941i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.50139 + 1.05628i\)
\(L(\frac12)\) \(\approx\) \(1.50139 + 1.05628i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.171 - 5.19i)T \)
7 \( 1 + (-11.7 - 14.3i)T \)
good2 \( 1 - 1.10iT - 8T^{2} \)
5 \( 1 + (-6.23 + 10.8i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-3.22 + 1.86i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (68.0 - 39.2i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-56.8 + 98.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-33.4 + 19.3i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (32.5 + 18.8i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (144. + 83.7i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 212. iT - 2.97e4T^{2} \)
37 \( 1 + (-132. - 228. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (190. + 329. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-90.2 + 156. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 154.T + 1.03e5T^{2} \)
53 \( 1 + (-162. - 93.5i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + 468.T + 2.05e5T^{2} \)
61 \( 1 + 356. iT - 2.26e5T^{2} \)
67 \( 1 + 96.7T + 3.00e5T^{2} \)
71 \( 1 - 705. iT - 3.57e5T^{2} \)
73 \( 1 + (631. + 364. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 409.T + 4.93e5T^{2} \)
83 \( 1 + (321. - 556. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-74.4 - 128. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-605. - 349. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−14.86877408377980152126951000294, −13.97841312321063140532097843776, −12.06586310584702594836929508383, −11.50522581373612997129754761310, −9.828921325444468077477200981136, −9.015929491286458418979918400736, −7.57850986054329381606517665062, −5.66971534127800985599663963464, −4.87401340570320986535787722032, −2.38658593485595264143876341024, 1.63181139912411999323770057408, 3.06629623898696120128454161417, 5.83549373374072060163685774972, 7.10854799688845262999276213873, 7.80965108783605329014350674137, 10.10861394170644309938306939759, 10.82626644052757815417172050288, 12.03461470854009183052948670208, 12.95726614364388117219489620521, 14.43095481858351727566071511070

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