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Properties

Label 2-882-63.25-c1-0-2
Degree $2$
Conductor $882$
Sign $-0.841 - 0.540i$
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 + 1.73i)5-s + (1.5 − 0.866i)6-s − 8-s + (1.5 − 2.59i)9-s + (1 − 1.73i)10-s + (−0.5 − 0.866i)11-s + (−1.5 + 0.866i)12-s + (3 + 5.19i)13-s − 3.46i·15-s + 16-s + (2.5 − 4.33i)17-s + (−1.5 + 2.59i)18-s + (3.5 + 6.06i)19-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.866 + 0.499i)3-s + 0.5·4-s + (−0.447 + 0.774i)5-s + (0.612 − 0.353i)6-s − 0.353·8-s + (0.5 − 0.866i)9-s + (0.316 − 0.547i)10-s + (−0.150 − 0.261i)11-s + (−0.433 + 0.249i)12-s + (0.832 + 1.44i)13-s − 0.894i·15-s + 0.250·16-s + (0.606 − 1.05i)17-s + (−0.353 + 0.612i)18-s + (0.802 + 1.39i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.841 - 0.540i$
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: \((2,\ 882,\ (\ :1/2),\ -0.841 - 0.540i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.156081 + 0.531385i\)
\(L(\frac12)\) \(\approx\) \(0.156081 + 0.531385i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3 - 5.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2 + 3.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 7T + 59T^{2} \)
61 \( 1 + 12T + 61T^{2} \)
67 \( 1 - 13T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 + (8 - 13.8i)T + (-41.5 - 71.8i)T^{2} \)
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

−10.54439297188163657083628063129, −9.648431661569956162718622482704, −9.103694258094576243803690039613, −7.79016093437395473020088930930, −7.13530196389849004217732764488, −6.22948970457344119710344931846, −5.44702397361330092355384382270, −4.04634267504930033625117684234, −3.24233093126644199620966741609, −1.43371243772335036104866388614, 0.42182174258186969404668511602, 1.48981932754384510933182340690, 3.16266743071336553936501861693, 4.64932235409531750128218172513, 5.51470822196765610585336651363, 6.37537081929873469219680608052, 7.40282300269220816088697540612, 8.117636053359188723921853900878, 8.732951102412849179973389690547, 9.952499250845921099928047023909

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