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Properties

Label 2-63-63.5-c3-0-18
Degree $2$
Conductor $63$
Sign $-0.989 + 0.142i$
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  − 1.81i·2-s + (−3.99 − 3.31i)3-s + 4.69·4-s + (−5.16 − 8.94i)5-s + (−6.03 + 7.26i)6-s + (−16.8 + 7.78i)7-s − 23.0i·8-s + (4.97 + 26.5i)9-s + (−16.2 + 9.38i)10-s + (−10.1 − 5.87i)11-s + (−18.7 − 15.5i)12-s + (−46.7 − 26.9i)13-s + (14.1 + 30.5i)14-s + (−9.03 + 52.8i)15-s − 4.40·16-s + (−31.9 − 55.3i)17-s + ⋯
L(s)  = 1  − 0.642i·2-s + (−0.769 − 0.638i)3-s + 0.586·4-s + (−0.461 − 0.799i)5-s + (−0.410 + 0.494i)6-s + (−0.907 + 0.420i)7-s − 1.01i·8-s + (0.184 + 0.982i)9-s + (−0.514 + 0.296i)10-s + (−0.278 − 0.161i)11-s + (−0.451 − 0.374i)12-s + (−0.997 − 0.575i)13-s + (0.270 + 0.583i)14-s + (−0.155 + 0.910i)15-s − 0.0687·16-s + (−0.456 − 0.790i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.989 + 0.142i$
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: \((2,\ 63,\ (\ :3/2),\ -0.989 + 0.142i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0612197 - 0.852258i\)
\(L(\frac12)\) \(\approx\) \(0.0612197 - 0.852258i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.99 + 3.31i)T \)
7 \( 1 + (16.8 - 7.78i)T \)
good2 \( 1 + 1.81iT - 8T^{2} \)
5 \( 1 + (5.16 + 8.94i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (10.1 + 5.87i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (46.7 + 26.9i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (31.9 + 55.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-87.6 - 50.6i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-168. + 97.3i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-13.4 + 7.77i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 200. iT - 2.97e4T^{2} \)
37 \( 1 + (152. - 264. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-35.3 + 61.2i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-52.4 - 90.9i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 7.27T + 1.03e5T^{2} \)
53 \( 1 + (460. - 266. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + 174.T + 2.05e5T^{2} \)
61 \( 1 - 301. iT - 2.26e5T^{2} \)
67 \( 1 - 298.T + 3.00e5T^{2} \)
71 \( 1 + 709. iT - 3.57e5T^{2} \)
73 \( 1 + (-732. + 423. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 - 1.13e3T + 4.93e5T^{2} \)
83 \( 1 + (-164. - 284. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (506. - 876. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-1.20e3 + 694. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−13.34170727213200135512642307105, −12.43338169361439416703854554323, −11.98628311063830073529623572468, −10.77075651205512751009547283286, −9.531604354559304189353170597854, −7.74305347339055230003556451838, −6.56217270100151630980277835039, −5.06641288477064207311053202689, −2.76615596741089459113831463133, −0.62277209056304601931926144861, 3.26562097220768953198208952668, 5.17944822762510927177974286360, 6.72956062679158136553437111380, 7.23244262580690612694610292552, 9.367122124211841215633743976986, 10.64878246845340976315020604733, 11.36491420444875467789368807117, 12.59445626429968640263559806220, 14.32610579993096523001641233054, 15.36279158549561761833389867616

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