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Properties

Label 2-882-9.7-c1-0-29
Degree $2$
Conductor $882$
Sign $-0.939 + 0.342i$
Analytic cond. $7.04280$
Root an. cond. $2.65382$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

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 − 2.59i)5-s − 1.73i·6-s + 0.999·8-s + (1.5 − 2.59i)9-s + 3·10-s + (1.5 − 2.59i)11-s + (1.49 + 0.866i)12-s + (2.5 + 4.33i)13-s + (4.5 + 2.59i)15-s + (−0.5 + 0.866i)16-s − 3·17-s + (1.5 + 2.59i)18-s − 5·19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.866 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.670 − 1.16i)5-s − 0.707i·6-s + 0.353·8-s + (0.5 − 0.866i)9-s + 0.948·10-s + (0.452 − 0.783i)11-s + (0.433 + 0.250i)12-s + (0.693 + 1.20i)13-s + (1.16 + 0.670i)15-s + (−0.125 + 0.216i)16-s − 0.727·17-s + (0.353 + 0.612i)18-s − 1.14·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.5 - 9.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.5 + 2.59i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.523756629809606465296259622115, −8.794416419967053015032500806870, −8.406572554738207186652410747970, −7.00337385015735356382566383571, −6.31161855661158165100669513573, −5.40240999221953605980898689631, −4.40176178283255397633132059669, −3.90816121421174497149503325854, −1.38710966815627223188826068273, 0, 1.73913013012394302611000660936, 3.01586743033391632798713887408, 4.08761638957052283837292033521, 5.20092372061562563148816864730, 6.63130700951211029382758832697, 6.88102877667591105738239412144, 7.973657625687070944670092759898, 8.724726512086160746890719913753, 10.19562869270034713479775873560

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