Properties

Label 2-882-3.2-c4-0-47
Degree $2$
Conductor $882$
Sign $0.816 + 0.577i$
Analytic cond. $91.1723$
Root an. cond. $9.54841$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82i·2-s − 8.00·4-s + 44.3i·5-s − 22.6i·8-s − 125.·10-s + 72.1i·11-s + 119.·13-s + 64.0·16-s − 166. i·17-s + 146.·19-s − 355. i·20-s − 204.·22-s − 586. i·23-s − 1.34e3·25-s + 337. i·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 + 0.596i·11-s + 0.705·13-s + 0.250·16-s − 0.577i·17-s + 0.404·19-s − 0.887i·20-s − 0.421·22-s − 1.10i·23-s − 2.15·25-s + 0.498i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(91.1723\)
Root analytic conductor: \(9.54841\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :2),\ 0.816 + 0.577i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.5381610235\)
\(L(\frac12)\) \(\approx\) \(0.5381610235\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82iT \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 44.3iT - 625T^{2} \)
11 \( 1 - 72.1iT - 1.46e4T^{2} \)
13 \( 1 - 119.T + 2.85e4T^{2} \)
17 \( 1 + 166. iT - 8.35e4T^{2} \)
19 \( 1 - 146.T + 1.30e5T^{2} \)
23 \( 1 + 586. iT - 2.79e5T^{2} \)
29 \( 1 + 1.11e3iT - 7.07e5T^{2} \)
31 \( 1 + 1.71e3T + 9.23e5T^{2} \)
37 \( 1 + 2.09e3T + 1.87e6T^{2} \)
41 \( 1 - 2.39e3iT - 2.82e6T^{2} \)
43 \( 1 - 324.T + 3.41e6T^{2} \)
47 \( 1 + 1.73e3iT - 4.87e6T^{2} \)
53 \( 1 + 2.80e3iT - 7.89e6T^{2} \)
59 \( 1 + 5.42e3iT - 1.21e7T^{2} \)
61 \( 1 - 1.58e3T + 1.38e7T^{2} \)
67 \( 1 + 1.05e3T + 2.01e7T^{2} \)
71 \( 1 - 4.24e3iT - 2.54e7T^{2} \)
73 \( 1 - 5.14e3T + 2.83e7T^{2} \)
79 \( 1 + 8.61e3T + 3.89e7T^{2} \)
83 \( 1 + 4.62e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.21e4iT - 6.27e7T^{2} \)
97 \( 1 + 9.81e3T + 8.85e7T^{2} \)
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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

−9.600118185978069627776682103591, −8.462467961220340893956151574221, −7.53548295104583498723349947606, −6.88692899124476202266812678201, −6.30040968023340763411715864467, −5.28446596370527800462947003914, −4.01203305538462660320411539580, −3.13952449386003280525644381667, −2.01168557291929513251639228129, −0.12703997842131300392351915024, 1.09983075448834068412875668396, 1.70638827132145696275185000888, 3.41609639179540997523674410498, 4.12469584136477636580123970932, 5.34068558953964078631759475391, 5.65833962967598071411653336015, 7.29698907275831142737826870940, 8.358775805328489851585701261473, 8.918105428733958463737411807580, 9.414528600615544028228513885920

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