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Properties

Label 2-882-9.4-c1-0-31
Degree $2$
Conductor $882$
Sign $0.999 + 0.0334i$
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 + (−0.619 + 1.61i)3-s + (−0.499 + 0.866i)4-s + (1.59 − 2.75i)5-s + (−1.71 + 0.272i)6-s − 0.999·8-s + (−2.23 − 2.00i)9-s + 3.18·10-s + (−1.59 − 2.75i)11-s + (−1.09 − 1.34i)12-s + (2.85 − 4.93i)13-s + (3.47 + 4.28i)15-s + (−0.5 − 0.866i)16-s + 1.52·17-s + (0.619 − 2.93i)18-s + 1.28·19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.357 + 0.933i)3-s + (−0.249 + 0.433i)4-s + (0.711 − 1.23i)5-s + (−0.698 + 0.111i)6-s − 0.353·8-s + (−0.744 − 0.668i)9-s + 1.00·10-s + (−0.479 − 0.830i)11-s + (−0.314 − 0.388i)12-s + (0.790 − 1.36i)13-s + (0.896 + 1.10i)15-s + (−0.125 − 0.216i)16-s + 0.369·17-s + (0.146 − 0.691i)18-s + 0.294·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.999 + 0.0334i$
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: \((2,\ 882,\ (\ :1/2),\ 0.999 + 0.0334i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64054 - 0.0274690i\)
\(L(\frac12)\) \(\approx\) \(1.64054 - 0.0274690i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.619 - 1.61i)T \)
7 \( 1 \)
good5 \( 1 + (-1.59 + 2.75i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.59 + 2.75i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.85 + 4.93i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.52T + 17T^{2} \)
19 \( 1 - 1.28T + 19T^{2} \)
23 \( 1 + (1.11 - 1.93i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.54 + 6.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.71 + 8.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (2.80 - 4.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.41 - 5.91i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.91 - 5.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.05T + 53T^{2} \)
59 \( 1 + (-0.562 + 0.974i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.56 + 2.70i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.48 - 9.49i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.69T + 71T^{2} \)
73 \( 1 - 4.96T + 73T^{2} \)
79 \( 1 + (-2.06 - 3.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.03 + 6.98i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 0.225T + 89T^{2} \)
97 \( 1 + (-7.42 - 12.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.893037140678974178999666177947, −9.372111570723994803297917025874, −8.346972004559955260556932598624, −7.917761781804951435000711570644, −6.01915913539053626085803436887, −5.80676516797965043836080470921, −5.02841739944039028659275330582, −4.04831195630895217181137727373, −2.96027653085674315588026649525, −0.76691125660843399200042336221, 1.60752441804350838460588554488, 2.37914616324586324900511781359, 3.48578198654754047254959318870, 4.92868627411304202468032820975, 5.87787895924820000099887063422, 6.76615867646802994682505534078, 7.16268394521650274742897968641, 8.513972313878062558754640963757, 9.493704905343100460333267478768, 10.58466747969075770155463034181

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