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Properties

Label 2-828-828.551-c0-0-1
Degree $2$
Conductor $828$
Sign $-0.642 - 0.766i$
Analytic cond. $0.413225$
Root an. cond. $0.642826$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.766 + 0.642i)2-s + (−0.173 + 0.984i)3-s + (0.173 + 0.984i)4-s + (−0.766 + 0.642i)6-s + (−0.500 + 0.866i)8-s + (−0.939 − 0.342i)9-s − 12-s + (0.173 + 0.300i)13-s + (−0.939 + 0.342i)16-s + (−0.499 − 0.866i)18-s + (0.5 + 0.866i)23-s + (−0.766 − 0.642i)24-s + (0.5 − 0.866i)25-s + (−0.0603 + 0.342i)26-s + (0.5 − 0.866i)27-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (−0.173 + 0.984i)3-s + (0.173 + 0.984i)4-s + (−0.766 + 0.642i)6-s + (−0.500 + 0.866i)8-s + (−0.939 − 0.342i)9-s − 12-s + (0.173 + 0.300i)13-s + (−0.939 + 0.342i)16-s + (−0.499 − 0.866i)18-s + (0.5 + 0.866i)23-s + (−0.766 − 0.642i)24-s + (0.5 − 0.866i)25-s + (−0.0603 + 0.342i)26-s + (0.5 − 0.866i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(828\)    =    \(2^{2} \cdot 3^{2} \cdot 23\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(0.413225\)
Root analytic conductor: \(0.642826\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{828} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 828,\ (\ :0),\ -0.642 - 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.317044340\)
\(L(\frac12)\) \(\approx\) \(1.317044340\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 - 0.642i)T \)
3 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
29 \( 1 + (1.11 + 0.642i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-1.11 + 0.642i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.592 - 0.342i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + 1.53T + T^{2} \)
73 \( 1 - 1.53T + T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−10.84301627630590627247247989004, −9.838400708371403124811970309752, −8.963464534041243951576987964916, −8.201040649596186616261522919641, −7.14800237200510219181085035981, −6.14954617675612973932550197552, −5.41700798224171864317942073396, −4.46934166250668053981089914056, −3.72909867107743577278975519155, −2.60402736017968118868529168791, 1.20988694900037861410300603997, 2.47828026962149973078690084363, 3.46048270903789115890519751619, 4.84493150939720709906537663411, 5.63939165119995228844032611275, 6.56888770391099370359994575443, 7.27768092090977965760304046040, 8.459621958688360666234568582793, 9.320488767935185855629589906490, 10.53030228219018871395380467955

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