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Properties

Label 2-882-9.4-c1-0-9
Degree $2$
Conductor $882$
Sign $-0.262 - 0.964i$
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 + (0.933 + 1.45i)3-s + (−0.499 + 0.866i)4-s + (−0.230 + 0.398i)5-s + (0.796 − 1.53i)6-s + 0.999·8-s + (−1.25 + 2.72i)9-s + 0.460·10-s + (1.82 + 3.15i)11-s + (−1.73 + 0.0789i)12-s + (−0.730 + 1.26i)13-s + (−0.796 + 0.0363i)15-s + (−0.5 − 0.866i)16-s − 3.73·17-s + (2.98 − 0.273i)18-s − 4.05·19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.538 + 0.842i)3-s + (−0.249 + 0.433i)4-s + (−0.102 + 0.178i)5-s + (0.325 − 0.627i)6-s + 0.353·8-s + (−0.419 + 0.907i)9-s + 0.145·10-s + (0.549 + 0.952i)11-s + (−0.499 + 0.0227i)12-s + (−0.202 + 0.350i)13-s + (−0.205 + 0.00938i)15-s + (−0.125 − 0.216i)16-s − 0.905·17-s + (0.704 − 0.0643i)18-s − 0.930·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.262 - 0.964i$
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: \((2,\ 882,\ (\ :1/2),\ -0.262 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.670558 + 0.877408i\)
\(L(\frac12)\) \(\approx\) \(0.670558 + 0.877408i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.933 - 1.45i)T \)
7 \( 1 \)
good5 \( 1 + (0.230 - 0.398i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.82 - 3.15i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.730 - 1.26i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.73T + 17T^{2} \)
19 \( 1 + 4.05T + 19T^{2} \)
23 \( 1 + (0.566 - 0.981i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.48 + 7.77i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.257 - 0.445i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.10T + 37T^{2} \)
41 \( 1 + (-0.472 + 0.819i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.66 - 8.07i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.16 - 2.01i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 12.4T + 53T^{2} \)
59 \( 1 + (6.44 - 11.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.04 - 10.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.16 + 2.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.67T + 71T^{2} \)
73 \( 1 + 13.2T + 73T^{2} \)
79 \( 1 + (-2.50 - 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.32 + 5.75i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.72T + 89T^{2} \)
97 \( 1 + (-5.59 - 9.68i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.29536545839241275359206481567, −9.420208052635147185605810747714, −9.123737236654428185013057852911, −8.024437178887636210874022040893, −7.24320818642201148974568093051, −6.02455957268061562223062968438, −4.48107074577826830496612455077, −4.20788426783118535793088224913, −2.85472356020979899481084827048, −1.90669855114734031407795588638, 0.54296962872212431414735931939, 2.02134690490939877612695741801, 3.35931554086640352305274923772, 4.57515989992684649891162188458, 5.91959236632380549449445389084, 6.52404120009143073811586823437, 7.36968797455707803002139284365, 8.320230295888336430410853280387, 8.755626878802998613516509744987, 9.517620052073149239717092062202

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