Properties

Label 2-882-7.2-c3-0-6
Degree $2$
Conductor $882$
Sign $-0.991 - 0.126i$
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 + 1.73i)2-s + (−1.99 − 3.46i)4-s + (−5.22 + 9.04i)5-s + 7.99·8-s + (−10.4 − 18.0i)10-s + (−30.5 − 52.9i)11-s + 59.2·13-s + (−8 + 13.8i)16-s + (10.2 + 17.7i)17-s + (−40.1 + 69.5i)19-s + 41.7·20-s + 122.·22-s + (79.1 − 137. i)23-s + (7.93 + 13.7i)25-s + (−59.2 + 102. i)26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.467 + 0.809i)5-s + 0.353·8-s + (−0.330 − 0.572i)10-s + (−0.837 − 1.45i)11-s + 1.26·13-s + (−0.125 + 0.216i)16-s + (0.145 + 0.252i)17-s + (−0.485 + 0.840i)19-s + 0.467·20-s + 1.18·22-s + (0.717 − 1.24i)23-s + (0.0635 + 0.109i)25-s + (−0.446 + 0.773i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.991 - 0.126i$
Analytic conductor: \(52.0396\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6863404268\)
\(L(\frac12)\) \(\approx\) \(0.6863404268\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (5.22 - 9.04i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (30.5 + 52.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 59.2T + 2.19e3T^{2} \)
17 \( 1 + (-10.2 - 17.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (40.1 - 69.5i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-79.1 + 137. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 85.1T + 2.43e4T^{2} \)
31 \( 1 + (-121. - 210. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (145. - 251. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 168T + 6.89e4T^{2} \)
43 \( 1 - 7.62T + 7.95e4T^{2} \)
47 \( 1 + (84.6 - 146. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-125. - 216. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (402. + 697. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-16.5 + 28.7i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-138. - 240. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 631.T + 3.57e5T^{2} \)
73 \( 1 + (384. + 665. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-209. + 362. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 761.T + 5.71e5T^{2} \)
89 \( 1 + (786. - 1.36e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.04e3T + 9.12e5T^{2} \)
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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

−10.56074820392561773769429251380, −9.001492270502736566574992466945, −8.354240758683671360369062642362, −7.78112821738884533244455838025, −6.58913927395881682346007454728, −6.13009673672508674707719343371, −5.05820013297250289599540558898, −3.70806217806283388848049930181, −2.91672180158096617827312958212, −1.14831987697699292890258770638, 0.23191399094423486143631332551, 1.44464485161264357297756313280, 2.61291499748378412180360482372, 3.92997883640711184495508414464, 4.66867651908309278526935639981, 5.64555839740390114981233787666, 7.08383712971509045926098722312, 7.76906692296531350624588059813, 8.673134572113198719152020687822, 9.309273737731035592443770095164

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