Properties

Label 2-882-3.2-c2-0-4
Degree $2$
Conductor $882$
Sign $0.816 - 0.577i$
Analytic cond. $24.0327$
Root an. cond. $4.90232$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s − 2.00·4-s − 8.89i·5-s + 2.82i·8-s − 12.5·10-s + 17.7i·11-s − 2.58·13-s + 4.00·16-s + 25.8i·17-s − 20·19-s + 17.7i·20-s + 25.1·22-s + 17.7i·23-s − 54.1·25-s + 3.65i·26-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.500·4-s − 1.77i·5-s + 0.353i·8-s − 1.25·10-s + 1.61i·11-s − 0.198·13-s + 0.250·16-s + 1.52i·17-s − 1.05·19-s + 0.889i·20-s + 1.14·22-s + 0.773i·23-s − 2.16·25-s + 0.140i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(24.0327\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1),\ 0.816 - 0.577i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8624167340\)
\(L(\frac12)\) \(\approx\) \(0.8624167340\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 8.89iT - 25T^{2} \)
11 \( 1 - 17.7iT - 121T^{2} \)
13 \( 1 + 2.58T + 169T^{2} \)
17 \( 1 - 25.8iT - 289T^{2} \)
19 \( 1 + 20T + 361T^{2} \)
23 \( 1 - 17.7iT - 529T^{2} \)
29 \( 1 - 11.9iT - 841T^{2} \)
31 \( 1 - 17.1T + 961T^{2} \)
37 \( 1 - 38T + 1.36e3T^{2} \)
41 \( 1 - 15.7iT - 1.68e3T^{2} \)
43 \( 1 + 43.4T + 1.84e3T^{2} \)
47 \( 1 - 16.9iT - 2.20e3T^{2} \)
53 \( 1 + 85.5iT - 2.80e3T^{2} \)
59 \( 1 + 1.64iT - 3.48e3T^{2} \)
61 \( 1 + 100.T + 3.72e3T^{2} \)
67 \( 1 - 36.6T + 4.48e3T^{2} \)
71 \( 1 - 17.7iT - 5.04e3T^{2} \)
73 \( 1 + 28.9T + 5.32e3T^{2} \)
79 \( 1 - 118.T + 6.24e3T^{2} \)
83 \( 1 - 120. iT - 6.88e3T^{2} \)
89 \( 1 - 139. iT - 7.92e3T^{2} \)
97 \( 1 + 44.4T + 9.40e3T^{2} \)
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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

−9.842635270816689860794496660884, −9.377791944693159306251872105370, −8.409080484206988633236237406237, −7.88504752118313328368449733275, −6.47302127890892796511219176598, −5.27579683939561764961659537325, −4.56808557060107565393695849427, −3.89632410405985058936055770376, −2.08873890762636267050547469465, −1.32421606524410398397416818128, 0.29130725124109142278209907130, 2.58594777894966329508181404296, 3.29645970813294176616998095564, 4.52631805254222033115541580933, 5.88854674261695690186399994311, 6.39581867388779198405957859918, 7.18894495424599368834722920403, 7.989160345681236192658396277988, 8.899172980072316951531512822956, 9.896466213157686488732588377237

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