Properties

Label 2-63-21.17-c3-0-7
Degree $2$
Conductor $63$
Sign $0.171 + 0.985i$
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  + (4.21 − 2.43i)2-s + (7.83 − 13.5i)4-s + (−6.38 − 11.0i)5-s + (2.53 + 18.3i)7-s − 37.3i·8-s + (−53.7 − 31.0i)10-s + (46.8 + 27.0i)11-s + 8.85i·13-s + (55.3 + 71.1i)14-s + (−28.1 − 48.7i)16-s + (34.4 − 59.6i)17-s + (−141. + 81.9i)19-s − 200.·20-s + 263.·22-s + (−81.3 + 46.9i)23-s + ⋯
L(s)  = 1  + (1.48 − 0.860i)2-s + (0.979 − 1.69i)4-s + (−0.570 − 0.988i)5-s + (0.137 + 0.990i)7-s − 1.64i·8-s + (−1.70 − 0.981i)10-s + (1.28 + 0.741i)11-s + 0.188i·13-s + (1.05 + 1.35i)14-s + (−0.439 − 0.760i)16-s + (0.491 − 0.851i)17-s + (−1.71 + 0.989i)19-s − 2.23·20-s + 2.55·22-s + (−0.737 + 0.425i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.171 + 0.985i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.171 + 0.985i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.171 + 0.985i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.18272 - 1.83550i\)
\(L(\frac12)\) \(\approx\) \(2.18272 - 1.83550i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-2.53 - 18.3i)T \)
good2 \( 1 + (-4.21 + 2.43i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (6.38 + 11.0i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-46.8 - 27.0i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 8.85iT - 2.19e3T^{2} \)
17 \( 1 + (-34.4 + 59.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (141. - 81.9i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (81.3 - 46.9i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 119. iT - 2.43e4T^{2} \)
31 \( 1 + (-85.6 - 49.4i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (47.0 + 81.5i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 259.T + 6.89e4T^{2} \)
43 \( 1 - 5.01T + 7.95e4T^{2} \)
47 \( 1 + (28.6 + 49.6i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (407. + 235. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-112. + 195. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-370. + 213. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (81.9 - 141. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 79.8iT - 3.57e5T^{2} \)
73 \( 1 + (-666. - 384. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (267. + 463. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 438.T + 5.71e5T^{2} \)
89 \( 1 + (12.8 + 22.2i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.38e3iT - 9.12e5T^{2} \)
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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

−14.15744469655120937216323511907, −12.72594477390541031755980216466, −12.13691529825015453884475199525, −11.55317747209019989617269822562, −9.824761946431844920186371696317, −8.434338667676001213118596401583, −6.28205686663389725346782029075, −4.91597369974548760666038570306, −3.90569740179097622457595372378, −1.88409262890720601702671405750, 3.41706807510489796740302718193, 4.33444738358739292978815789527, 6.28763988621764045184495642579, 6.95583755071036649935948270825, 8.267412205079362291761110560058, 10.58948760480805888956697347150, 11.57050900028295449518291929521, 12.81984358894126677590719789333, 13.95159208159220068606508700549, 14.61045430795793707884336217667

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