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Properties

Label 2-882-7.4-c3-0-11
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 + (7.91 + 13.7i)5-s − 7.99·8-s + (−15.8 + 27.4i)10-s + (25.9 − 44.8i)11-s − 38.8·13-s + (−8 − 13.8i)16-s + (−13.6 + 23.6i)17-s + (38.2 + 66.2i)19-s − 63.3·20-s + 103.·22-s + (73.6 + 127. i)23-s + (−62.9 + 108. i)25-s + (−38.8 − 67.2i)26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.708 + 1.22i)5-s − 0.353·8-s + (−0.500 + 0.867i)10-s + (0.710 − 1.23i)11-s − 0.828·13-s + (−0.125 − 0.216i)16-s + (−0.195 + 0.337i)17-s + (0.461 + 0.800i)19-s − 0.708·20-s + 1.00·22-s + (0.667 + 1.15i)23-s + (−0.503 + 0.871i)25-s + (−0.292 − 0.507i)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: $\chi_{882} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.811700357\)
\(L(\frac12)\) \(\approx\) \(1.811700357\)
\(L(\frac{5}{2})\) 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 \( 1 + (-7.91 - 13.7i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-25.9 + 44.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 38.8T + 2.19e3T^{2} \)
17 \( 1 + (13.6 - 23.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-38.2 - 66.2i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-73.6 - 127. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 240.T + 2.43e4T^{2} \)
31 \( 1 + (148. - 256. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-80.7 - 139. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 102.T + 6.89e4T^{2} \)
43 \( 1 + 328.T + 7.95e4T^{2} \)
47 \( 1 + (33.9 + 58.8i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (33.2 - 57.5i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-230. + 400. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-92.6 - 160. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (272. - 472. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 130.T + 3.57e5T^{2} \)
73 \( 1 + (-90.6 + 157. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-204. - 354. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 347.T + 5.71e5T^{2} \)
89 \( 1 + (578. + 1.00e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.61e3T + 9.12e5T^{2} \)
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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

−10.09629180692918321376405527170, −9.367531232958033136701021714769, −8.427256015405939266627181010758, −7.33942303728178983169242465679, −6.75719186558209215686050713171, −5.88371795828572868664824978591, −5.24123556847711730260899033449, −3.66142747260451674768901084296, −3.08918601665691151542842637928, −1.65212421980913176842836742209, 0.38892006346671671723045187364, 1.64344715268488031470144209665, 2.45753723302696429280019665467, 4.03944049474429555707025198139, 4.84436891232929639537566900991, 5.41727240921048414782676705073, 6.65171624209624702610408500400, 7.57405439288665438556041731081, 8.927987244780318571584105993794, 9.402348028448734765795702341536

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