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Properties

Label 2-882-63.16-c1-0-15
Degree $2$
Conductor $882$
Sign $0.105 + 0.994i$
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 + (0.349 + 1.69i)3-s + (−0.499 + 0.866i)4-s − 3.69·5-s + (1.29 − 1.15i)6-s + 0.999·8-s + (−2.75 + 1.18i)9-s + (1.84 + 3.20i)10-s − 1.47·11-s + (−1.64 − 0.545i)12-s + (1.34 + 2.33i)13-s + (−1.29 − 6.27i)15-s + (−0.5 − 0.866i)16-s + (−3.28 − 5.69i)17-s + (2.40 + 1.79i)18-s + (0.444 − 0.769i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.201 + 0.979i)3-s + (−0.249 + 0.433i)4-s − 1.65·5-s + (0.528 − 0.469i)6-s + 0.353·8-s + (−0.918 + 0.395i)9-s + (0.584 + 1.01i)10-s − 0.445·11-s + (−0.474 − 0.157i)12-s + (0.374 + 0.648i)13-s + (−0.334 − 1.62i)15-s + (−0.125 − 0.216i)16-s + (−0.797 − 1.38i)17-s + (0.566 + 0.422i)18-s + (0.101 − 0.176i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.105 + 0.994i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.105 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.393353 - 0.353919i\)
\(L(\frac12)\) \(\approx\) \(0.393353 - 0.353919i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.349 - 1.69i)T \)
7 \( 1 \)
good5 \( 1 + 3.69T + 5T^{2} \)
11 \( 1 + 1.47T + 11T^{2} \)
13 \( 1 + (-1.34 - 2.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.28 + 5.69i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.444 + 0.769i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 6.28T + 23T^{2} \)
29 \( 1 + (-1.25 + 2.17i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.40 + 5.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.38 - 2.40i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.05 - 3.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.00618 + 0.0107i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.49 + 6.05i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.60 + 2.78i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.45 + 5.98i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.86 + 4.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.73 + 8.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.46T + 71T^{2} \)
73 \( 1 + (-6.03 - 10.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.72 + 9.91i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.23 - 3.87i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.43 + 7.68i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.58 + 11.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.914022886927405518531009348649, −9.131850002937741128770107086744, −8.437065181079716618778883521024, −7.70152529228599840352253621003, −6.73932158769527601795334425372, −4.97799916140919826475581289185, −4.42659095115916962199122121819, −3.48792214310596802882947827575, −2.64339038750555552170518651663, −0.33602037307279294487986369816, 1.11203542182934584380599048891, 2.92332368580282154704752283112, 3.98230970520837817897675783341, 5.19285808532713905336542272583, 6.34169763381382009073736607423, 7.12172384444572712288304201687, 7.80923242654595105303337436982, 8.435860508768538897893589983370, 8.933936896224472751734331252135, 10.58119073783996658411962792470

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