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Properties

Label 2-882-9.7-c1-0-31
Degree $2$
Conductor $882$
Sign $0.230 + 0.973i$
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 + (1.64 + 0.545i)3-s + (−0.499 − 0.866i)4-s + (−1.84 − 3.20i)5-s + (−1.29 + 1.15i)6-s + 0.999·8-s + (2.40 + 1.79i)9-s + 3.69·10-s + (0.738 − 1.27i)11-s + (−0.349 − 1.69i)12-s + (−1.34 − 2.33i)13-s + (−1.29 − 6.27i)15-s + (−0.5 + 0.866i)16-s − 6.57·17-s + (−2.75 + 1.18i)18-s + 0.888·19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.949 + 0.314i)3-s + (−0.249 − 0.433i)4-s + (−0.827 − 1.43i)5-s + (−0.528 + 0.469i)6-s + 0.353·8-s + (0.801 + 0.597i)9-s + 1.16·10-s + (0.222 − 0.385i)11-s + (−0.100 − 0.489i)12-s + (−0.374 − 0.648i)13-s + (−0.334 − 1.62i)15-s + (−0.125 + 0.216i)16-s − 1.59·17-s + (−0.649 + 0.279i)18-s + 0.203·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.230 + 0.973i$
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: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.230 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.858757 - 0.679376i\)
\(L(\frac12)\) \(\approx\) \(0.858757 - 0.679376i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 - 0.545i)T \)
7 \( 1 \)
good5 \( 1 + (1.84 + 3.20i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.738 + 1.27i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.34 + 2.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.57T + 17T^{2} \)
19 \( 1 - 0.888T + 19T^{2} \)
23 \( 1 + (3.14 + 5.44i)T + (-11.5 + 19.9i)T^{2} \)
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 \( 1 - 2.77T + 37T^{2} \)
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 \( 1 - 3.21T + 53T^{2} \)
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 \( 1 - 12.0T + 73T^{2} \)
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 \( 1 - 8.87T + 89T^{2} \)
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

−9.565350476708772800949829846264, −8.901880395101820761111309970971, −8.317167546720465942662959849103, −7.80568455055579258237156902609, −6.74405643092301298357708193457, −5.41106245426670838386304636911, −4.49913539319913577123014832801, −3.88565878420189518526940675817, −2.22877211727421014800527144491, −0.51438357022462864683837619678, 1.85636972395340665359023366157, 2.79883516029515238132778808629, 3.67282116554069470017345980651, 4.47469187905739501629859856463, 6.48037251691668793318401024396, 7.15699818699759807620831666098, 7.70424099006875866345114588008, 8.737562127377979650466077339265, 9.454656413939118857620304031807, 10.31619977163831816237185598945

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