Properties

Label 2-882-7.4-c3-0-33
Degree $2$
Conductor $882$
Sign $0.749 + 0.661i$
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 + (−2.29 − 3.97i)5-s − 7.99·8-s + (4.58 − 7.94i)10-s + (−3.24 + 5.61i)11-s − 45.2·13-s + (−8 − 13.8i)16-s + (−40.7 + 70.6i)17-s + (−2.52 − 4.37i)19-s + 18.3·20-s − 12.9·22-s + (53.1 + 92.0i)23-s + (51.9 − 90.0i)25-s + (−45.2 − 78.3i)26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.205 − 0.355i)5-s − 0.353·8-s + (0.145 − 0.251i)10-s + (−0.0888 + 0.153i)11-s − 0.964·13-s + (−0.125 − 0.216i)16-s + (−0.581 + 1.00i)17-s + (−0.0305 − 0.0528i)19-s + 0.205·20-s − 0.125·22-s + (0.481 + 0.834i)23-s + (0.415 − 0.720i)25-s + (−0.341 − 0.590i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.749 + 0.661i$
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: \((2,\ 882,\ (\ :3/2),\ 0.749 + 0.661i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.394014384\)
\(L(\frac12)\) \(\approx\) \(1.394014384\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.29 + 3.97i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (3.24 - 5.61i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 45.2T + 2.19e3T^{2} \)
17 \( 1 + (40.7 - 70.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (2.52 + 4.37i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-53.1 - 92.0i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 268.T + 2.43e4T^{2} \)
31 \( 1 + (-146. + 253. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (57.2 + 99.2i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 161.T + 6.89e4T^{2} \)
43 \( 1 + 471.T + 7.95e4T^{2} \)
47 \( 1 + (173. + 299. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-202. + 351. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-126. + 219. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (375. + 650. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (5.82 - 10.0i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 681.T + 3.57e5T^{2} \)
73 \( 1 + (-342. + 593. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (0.132 + 0.228i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 437.T + 5.71e5T^{2} \)
89 \( 1 + (-29.2 - 50.6i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.28e3T + 9.12e5T^{2} \)
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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

−9.627599008287067019858286065858, −8.506448691744356321990688754380, −8.055788958127365801970805526399, −6.95281073799891623302048060598, −6.30155404260583660587168673293, −5.09570395837229049823408273628, −4.52657336495654746081666719398, −3.38331683814579567936102602662, −2.08952014264613264108021015365, −0.36127376845765780076231228750, 1.04836784094789635516759232681, 2.57845861339206173677036291346, 3.17223837775400946580165782771, 4.63050279976944092648708099988, 5.06357129771516467069248785601, 6.52117307653332419221556341967, 7.08128515976452681167555377414, 8.315480264734920342970561557258, 9.091117290393997548983008528745, 10.12228599220177778800562557217

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