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Properties

Label 2-882-63.58-c1-0-19
Degree $2$
Conductor $882$
Sign $0.888 + 0.458i$
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 + (−0.5 − 0.866i)5-s + (−1.5 + 0.866i)6-s + 8-s + (1.5 − 2.59i)9-s + (−0.5 − 0.866i)10-s + (1 − 1.73i)11-s + (−1.5 + 0.866i)12-s + (−1 + 1.73i)13-s + (1.5 + 0.866i)15-s + 16-s + (1.5 − 2.59i)18-s + (3.5 − 6.06i)19-s + (−0.5 − 0.866i)20-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.866 + 0.499i)3-s + 0.5·4-s + (−0.223 − 0.387i)5-s + (−0.612 + 0.353i)6-s + 0.353·8-s + (0.5 − 0.866i)9-s + (−0.158 − 0.273i)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 + 0.250·16-s + (0.353 − 0.612i)18-s + (0.802 − 1.39i)19-s + (−0.111 − 0.193i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.888 + 0.458i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.888 + 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70817 - 0.414232i\)
\(L(\frac12)\) \(\approx\) \(1.70817 - 0.414232i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
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 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
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 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-3 + 5.19i)T + (-18.5 - 32.0i)T^{2} \)
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 \( 1 + 8T + 47T^{2} \)
53 \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 13T + 61T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 + 5T + 71T^{2} \)
73 \( 1 + (-7 - 12.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 11T + 79T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7 - 12.1i)T + (-44.5 - 77.0i)T^{2} \)
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

−10.25396209579769943462072331811, −9.326112427445416328035364320842, −8.495074317128414875875664217310, −7.13999089909872014995369911906, −6.51498100374461079882812658438, −5.50351430442072685595884490527, −4.73613154969714712791934883050, −4.02514432846578855863203501648, −2.77221069967928847209810332960, −0.865013661055320781129007301315, 1.34448496955604964700859277713, 2.74718937281670050114318812260, 4.00724754078071634669650158367, 4.98579465736579177885288479322, 5.87874626957330715393465897126, 6.58060744470450375492320112795, 7.53366370039563382079210199850, 8.051663524382363805446190205549, 9.787730416145806689417618209518, 10.24842505073903847604436003636

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