Properties

Label 2-882-9.4-c1-0-3
Degree $2$
Conductor $882$
Sign $0.766 - 0.642i$
Analytic cond. $7.04280$
Root an. cond. $2.65382$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s − 1.73i·3-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−1.49 + 0.866i)6-s + 0.999·8-s − 2.99·9-s + 0.999·10-s + (1 + 1.73i)11-s + (1.49 + 0.866i)12-s + (−1 + 1.73i)13-s + (1.49 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (1.49 + 2.59i)18-s − 7·19-s + (−0.499 − 0.866i)20-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s − 0.999i·3-s + (−0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s + (−0.612 + 0.353i)6-s + 0.353·8-s − 0.999·9-s + 0.316·10-s + (0.301 + 0.522i)11-s + (0.433 + 0.249i)12-s + (−0.277 + 0.480i)13-s + (0.387 + 0.223i)15-s + (−0.125 − 0.216i)16-s + (0.353 + 0.612i)18-s − 1.60·19-s + (−0.111 − 0.193i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.624276 + 0.227217i\)
\(L(\frac12)\) \(\approx\) \(0.624276 + 0.227217i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + 1.73iT \)
7 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4 - 6.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (-6 + 10.3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.5 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5T + 71T^{2} \)
73 \( 1 + 14T + 73T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.51696023103365853327884837393, −9.172582569827637244272936447534, −8.723106176718281520329632477332, −7.57317342490977563656494597802, −7.04055208374458847289762896191, −6.13943128522560623072417552031, −4.80737181645318555065345541655, −3.60009953889248425198930802688, −2.46001900906047846564147384262, −1.47366099525737588514248138513, 0.36283678923849524452357290607, 2.54644472288887540353028197812, 4.02866685332688906104050863501, 4.62247828969651111851188165476, 5.76679722819547495800601514875, 6.40291338645449863062726797109, 7.73807044408705986066664800566, 8.586182597597517043864144366873, 8.938362876323731976898734086150, 10.13833456294304709366078469746

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