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Properties

Label 2-882-21.17-c3-0-12
Degree $2$
Conductor $882$
Sign $0.987 + 0.154i$
Analytic cond. $52.0396$
Root an. cond. $7.21385$
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  + (1.73 − i)2-s + (1.99 − 3.46i)4-s + (−7.15 − 12.3i)5-s − 7.99i·8-s + (−24.7 − 14.3i)10-s + (51.7 + 29.8i)11-s + 53.7i·13-s + (−8 − 13.8i)16-s + (−67.8 + 117. i)17-s + (71.3 − 41.2i)19-s − 57.2·20-s + 119.·22-s + (18.0 − 10.3i)23-s + (−39.9 + 69.1i)25-s + (53.7 + 93.1i)26-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.640 − 1.10i)5-s − 0.353i·8-s + (−0.783 − 0.452i)10-s + (1.41 + 0.818i)11-s + 1.14i·13-s + (−0.125 − 0.216i)16-s + (−0.968 + 1.67i)17-s + (0.861 − 0.497i)19-s − 0.640·20-s + 1.15·22-s + (0.163 − 0.0942i)23-s + (−0.319 + 0.553i)25-s + (0.405 + 0.702i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.154i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.987 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.987 + 0.154i$
Analytic conductor: \(52.0396\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ 0.987 + 0.154i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.723435453\)
\(L(\frac12)\) \(\approx\) \(2.723435453\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.73 + i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (7.15 + 12.3i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-51.7 - 29.8i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 53.7iT - 2.19e3T^{2} \)
17 \( 1 + (67.8 - 117. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-71.3 + 41.2i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-18.0 + 10.3i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 63.0iT - 2.43e4T^{2} \)
31 \( 1 + (-105. - 60.7i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-162. - 281. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 206.T + 6.89e4T^{2} \)
43 \( 1 - 274.T + 7.95e4T^{2} \)
47 \( 1 + (147. + 254. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-154. - 89.4i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-85.0 + 147. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (63.0 - 36.4i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-526. + 912. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 28.8iT - 3.57e5T^{2} \)
73 \( 1 + (417. + 241. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-353. - 612. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 778.T + 5.71e5T^{2} \)
89 \( 1 + (-4.94 - 8.56i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.56e3iT - 9.12e5T^{2} \)
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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

−9.586600663309164951829894072875, −8.993156016320144683981390888230, −8.199364415370743057679293040986, −6.90102169491263279010109293894, −6.34547701182549677537620681117, −4.90383336350301694835720486978, −4.36834767784569714173666236538, −3.65122869691100334868395296988, −1.94720142502383071907400322117, −1.08121655002919533514579776424, 0.68322577368548002012798258091, 2.66823071438169687866561508261, 3.38106426837868241891613999979, 4.25666489140821583386205807955, 5.49915634004677560510004483380, 6.36321976789551043868977809648, 7.15291477391240426148783523650, 7.76604191462227696064484200147, 8.844268080354645222068620364107, 9.762437152442394351037949300050

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