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Properties

Label 2-63-9.4-c3-0-8
Degree $2$
Conductor $63$
Sign $0.939 + 0.342i$
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  + (0.5 + 0.866i)2-s + (−4.5 + 2.59i)3-s + (3.5 − 6.06i)4-s + (7 − 12.1i)5-s + (−4.5 − 2.59i)6-s + (3.5 + 6.06i)7-s + 15·8-s + (13.5 − 23.3i)9-s + 14·10-s + (23.5 + 40.7i)11-s + 36.3i·12-s + (43 − 74.4i)13-s + (−3.5 + 6.06i)14-s + 72.7i·15-s + (−20.5 − 35.5i)16-s − 9·17-s + ⋯
L(s)  = 1  + (0.176 + 0.306i)2-s + (−0.866 + 0.499i)3-s + (0.437 − 0.757i)4-s + (0.626 − 1.08i)5-s + (−0.306 − 0.176i)6-s + (0.188 + 0.327i)7-s + 0.662·8-s + (0.5 − 0.866i)9-s + 0.442·10-s + (0.644 + 1.11i)11-s + 0.874i·12-s + (0.917 − 1.58i)13-s + (−0.0668 + 0.115i)14-s + 1.25i·15-s + (−0.320 − 0.554i)16-s − 0.128·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.49467 - 0.263551i\)
\(L(\frac12)\) \(\approx\) \(1.49467 - 0.263551i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.5 - 2.59i)T \)
7 \( 1 + (-3.5 - 6.06i)T \)
good2 \( 1 + (-0.5 - 0.866i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-7 + 12.1i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-23.5 - 40.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-43 + 74.4i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 9T + 4.91e3T^{2} \)
19 \( 1 + 131T + 6.85e3T^{2} \)
23 \( 1 + (-6 + 10.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-130 - 225. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-27 + 46.7i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 246T + 5.06e4T^{2} \)
41 \( 1 + (191.5 - 331. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-84.5 - 146. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (48 + 83.1i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 300T + 1.48e5T^{2} \)
59 \( 1 + (214.5 - 371. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-190 - 329. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-77.5 + 134. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 72T + 3.57e5T^{2} \)
73 \( 1 - 117T + 3.89e5T^{2} \)
79 \( 1 + (-263 - 455. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (288 + 498. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 278T + 7.04e5T^{2} \)
97 \( 1 + (-100.5 - 174. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−14.80125377606123744543271827273, −13.12085673006724333159895646058, −12.26029805096234377004496723780, −10.81656285649550226193067705309, −9.997169268237818240899396228131, −8.707039851383544013977005202421, −6.62047489362043135800531936409, −5.57316288467973718676690166888, −4.64982512491707940059074044126, −1.32510133324512315078229015121, 2.01892294775176455883937657578, 3.99163577207341342416768036240, 6.34405943293118182340417073695, 6.79480843933665008305525235060, 8.495237847864993181146859867922, 10.55977609410320100580296240607, 11.18357195803922782349062038592, 12.04212091142202695105374816918, 13.52505723641251031401071115219, 14.02910019829895486564225608619

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