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Properties

Label 2-63-7.2-c3-0-5
Degree $2$
Conductor $63$
Sign $0.998 + 0.0589i$
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  + (−2.27 + 3.94i)2-s + (−6.38 − 11.0i)4-s + (8.93 − 15.4i)5-s + (2.26 − 18.3i)7-s + 21.6·8-s + (40.7 + 70.5i)10-s + (−5.69 − 9.86i)11-s − 13.0·13-s + (67.3 + 50.7i)14-s + (1.62 − 2.81i)16-s + (26.6 + 46.1i)17-s + (21.2 − 36.7i)19-s − 228.·20-s + 51.9·22-s + (76.0 − 131. i)23-s + ⋯
L(s)  = 1  + (−0.805 + 1.39i)2-s + (−0.797 − 1.38i)4-s + (0.799 − 1.38i)5-s + (0.122 − 0.992i)7-s + 0.958·8-s + (1.28 + 2.23i)10-s + (−0.156 − 0.270i)11-s − 0.279·13-s + (1.28 + 0.969i)14-s + (0.0254 − 0.0440i)16-s + (0.379 + 0.658i)17-s + (0.256 − 0.443i)19-s − 2.54·20-s + 0.503·22-s + (0.689 − 1.19i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0589i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.998 + 0.0589i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.998 + 0.0589i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.940238 - 0.0277375i\)
\(L(\frac12)\) \(\approx\) \(0.940238 - 0.0277375i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.26 + 18.3i)T \)
good2 \( 1 + (2.27 - 3.94i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (-8.93 + 15.4i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (5.69 + 9.86i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 13.0T + 2.19e3T^{2} \)
17 \( 1 + (-26.6 - 46.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-21.2 + 36.7i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-76.0 + 131. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 186.T + 2.43e4T^{2} \)
31 \( 1 + (-78.9 - 136. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (1.87 - 3.24i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 39.3T + 6.89e4T^{2} \)
43 \( 1 - 429.T + 7.95e4T^{2} \)
47 \( 1 + (-10.5 + 18.3i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-182. - 316. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (113. + 196. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (325. - 564. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (72.7 + 125. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 368.T + 3.57e5T^{2} \)
73 \( 1 + (304. + 527. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (455. - 788. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 327.T + 5.71e5T^{2} \)
89 \( 1 + (18.8 - 32.5i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 722.T + 9.12e5T^{2} \)
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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

−14.61071383177425856229227770748, −13.61925786538309071177007065584, −12.56184080953166616979702584993, −10.52422298935399090505955077206, −9.377797161277699072780685985864, −8.520490734209992736366240154171, −7.35684776850756491610247421500, −5.95496696203091485497266758813, −4.77168194223381093535637212297, −0.890307964620169678509682771276, 2.07171678858862665835590030403, 3.13314450190814193489436178205, 5.78240048439006285499258704692, 7.53462422835805311758776901408, 9.254774889089128561537358564826, 9.898954231681794175409365946620, 11.02686941924482691672476169850, 11.81148373018438271519525626985, 13.06230375374035091531553921533, 14.38030777033215778182919738886

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