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Properties

Label 2-882-9.4-c1-0-29
Degree $2$
Conductor $882$
Sign $0.173 + 0.984i$
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 + (1.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−1.5 − 0.866i)6-s + 0.999·8-s + (1.5 − 2.59i)9-s + (1.5 + 2.59i)11-s + 1.73i·12-s + (1 − 1.73i)13-s + (−0.5 − 0.866i)16-s + 3·17-s − 3·18-s + 19-s + (1.5 − 2.59i)22-s + (3 − 5.19i)23-s + (1.49 − 0.866i)24-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.866 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.612 − 0.353i)6-s + 0.353·8-s + (0.5 − 0.866i)9-s + (0.452 + 0.783i)11-s + 0.499i·12-s + (0.277 − 0.480i)13-s + (−0.125 − 0.216i)16-s + 0.727·17-s − 0.707·18-s + 0.229·19-s + (0.319 − 0.553i)22-s + (0.625 − 1.08i)23-s + (0.306 − 0.176i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.173 + 0.984i$
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: \((2,\ 882,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40452 - 1.17853i\)
\(L(\frac12)\) \(\approx\) \(1.40452 - 1.17853i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T \)
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 \( 1 - 3T + 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
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 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
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 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
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

−9.903032274368527798652125197368, −9.040276262268418094105582611928, −8.487544419431382483186701733064, −7.44059151717152319048962369091, −6.92928032823744119478600769676, −5.53762830792010891679050044600, −4.19667755000963154648833107927, −3.29962880189108273672831164416, −2.25798378495195265722827178245, −1.08028594500863224715436795326, 1.42891684345548479434657585203, 3.03926261833586846542103257069, 3.96481883176869853734896227193, 5.07335911007181134607128627244, 6.02555638910057487605016428280, 7.13658052134032557412985686404, 7.85634586241254813168047375829, 8.789105179004258680002390756483, 9.210979260704216312337666622340, 10.08638922281481078873384580728

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