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Properties

Label 2-882-7.2-c3-0-25
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 + (3 − 5.19i)5-s + 7.99·8-s + (6 + 10.3i)10-s + (−15 − 25.9i)11-s − 53·13-s + (−8 + 13.8i)16-s + (42 + 72.7i)17-s + (−48.5 + 84.0i)19-s − 24·20-s + 60·22-s + (42 − 72.7i)23-s + (44.5 + 77.0i)25-s + (53 − 91.7i)26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.268 − 0.464i)5-s + 0.353·8-s + (0.189 + 0.328i)10-s + (−0.411 − 0.712i)11-s − 1.13·13-s + (−0.125 + 0.216i)16-s + (0.599 + 1.03i)17-s + (−0.585 + 1.01i)19-s − 0.268·20-s + 0.581·22-s + (0.380 − 0.659i)23-s + (0.355 + 0.616i)25-s + (0.399 − 0.692i)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

\(L(2)\) \(\approx\) \(1.419622509\)
\(L(\frac12)\) \(\approx\) \(1.419622509\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3 + 5.19i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (15 + 25.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 53T + 2.19e3T^{2} \)
17 \( 1 + (-42 - 72.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (48.5 - 84.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-42 + 72.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 180T + 2.43e4T^{2} \)
31 \( 1 + (-89.5 - 155. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-72.5 + 125. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 126T + 6.89e4T^{2} \)
43 \( 1 + 325T + 7.95e4T^{2} \)
47 \( 1 + (-183 + 316. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (384 + 665. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-132 - 228. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-409 + 708. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-261.5 - 452. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 342T + 3.57e5T^{2} \)
73 \( 1 + (21.5 + 37.2i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-585.5 + 1.01e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 810T + 5.71e5T^{2} \)
89 \( 1 + (-300 + 519. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 386T + 9.12e5T^{2} \)
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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

−9.823579701990239421355464197369, −8.500020955045673934973615890006, −8.354584522892701799115657533375, −7.19014298159028862334067531642, −6.28542715394234372574205124823, −5.42596306802656806337613276836, −4.65856801786842666626219859070, −3.31987311241712133345987925301, −1.88762298925123600881300149651, −0.57501820450557357591991923647, 0.817551161758007230210496891808, 2.41969051922330492129602437876, 2.85477868235839737366656932100, 4.44208240083751175276253115551, 5.11042506835334813545510859284, 6.52075572437335769202827385101, 7.31363511003180748540107930928, 8.060511558356104849808430341502, 9.250281953240399211949457615188, 9.811059772996028032671376095765

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