Properties

Label 2-882-63.25-c1-0-13
Degree $2$
Conductor $882$
Sign $0.415 - 0.909i$
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  + 2-s + (−1.5 + 0.866i)3-s + 4-s + (−1.5 + 0.866i)6-s + 8-s + (1.5 − 2.59i)9-s + (1.5 + 2.59i)11-s + (−1.5 + 0.866i)12-s + (1 + 1.73i)13-s + 16-s + (−1.5 + 2.59i)17-s + (1.5 − 2.59i)18-s + (−0.5 − 0.866i)19-s + (1.5 + 2.59i)22-s + (3 − 5.19i)23-s + (−1.5 + 0.866i)24-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.866 + 0.499i)3-s + 0.5·4-s + (−0.612 + 0.353i)6-s + 0.353·8-s + (0.5 − 0.866i)9-s + (0.452 + 0.783i)11-s + (−0.433 + 0.249i)12-s + (0.277 + 0.480i)13-s + 0.250·16-s + (−0.363 + 0.630i)17-s + (0.353 − 0.612i)18-s + (−0.114 − 0.198i)19-s + (0.319 + 0.553i)22-s + (0.625 − 1.08i)23-s + (−0.306 + 0.176i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.415 - 0.909i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (655, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.415 - 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54223 + 0.991532i\)
\(L(\frac12)\) \(\approx\) \(1.54223 + 0.991532i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (1.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 3T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 - 5T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.63398598840422056134056481193, −9.540121927072017017045953848592, −8.818830527036998658770187764646, −7.38722286439270442099751683548, −6.59556751425781552594576285173, −5.95517297155671840314384534602, −4.72786586283657153955883377029, −4.35565461726357122346453008633, −3.10340554682511071615559895100, −1.46528985860581490841617847330, 0.867176545324441802531299364226, 2.39133924907586501260040783902, 3.70522992629002224772216777841, 4.78382455148093590420507382881, 5.70631654396046853779337353171, 6.30050962512421402604006492668, 7.22626784407767569365108821031, 8.033700956773000542445428960039, 9.169474520333243714809892661978, 10.29930591956881591662730591495

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