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Properties

Label 2-1188-33.20-c0-0-0
Degree $2$
Conductor $1188$
Sign $0.624 + 0.781i$
Analytic cond. $0.592889$
Root an. cond. $0.769993$
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  + (−1.53 + 0.5i)5-s + (−0.809 − 0.587i)7-s + (0.951 + 0.309i)11-s + (0.5 − 1.53i)13-s + (0.587 − 0.190i)17-s + (0.809 − 0.587i)19-s − 0.618i·23-s + (1.30 − 0.951i)25-s + (0.587 − 0.809i)29-s + (−0.309 + 0.951i)31-s + (1.53 + 0.5i)35-s + (−0.587 − 0.809i)41-s + (−0.951 − 1.30i)47-s + (0.951 + 0.309i)53-s − 1.61·55-s + ⋯
L(s)  = 1  + (−1.53 + 0.5i)5-s + (−0.809 − 0.587i)7-s + (0.951 + 0.309i)11-s + (0.5 − 1.53i)13-s + (0.587 − 0.190i)17-s + (0.809 − 0.587i)19-s − 0.618i·23-s + (1.30 − 0.951i)25-s + (0.587 − 0.809i)29-s + (−0.309 + 0.951i)31-s + (1.53 + 0.5i)35-s + (−0.587 − 0.809i)41-s + (−0.951 − 1.30i)47-s + (0.951 + 0.309i)53-s − 1.61·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1188 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1188 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1188\)    =    \(2^{2} \cdot 3^{3} \cdot 11\)
Sign: $0.624 + 0.781i$
Analytic conductor: \(0.592889\)
Root analytic conductor: \(0.769993\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1188} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1188,\ (\ :0),\ 0.624 + 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7386933149\)
\(L(\frac12)\) \(\approx\) \(0.7386933149\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 + (-0.951 - 0.309i)T \)
good5 \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \)
7 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + 0.618iT - T^{2} \)
29 \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 - 0.618T + T^{2} \)
71 \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \)
89 \( 1 + 1.61iT - T^{2} \)
97 \( 1 + (-0.809 - 0.587i)T^{2} \)
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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

−10.15835072572137579480671242483, −8.885289676711786044137978049723, −8.139576873756280143463694273806, −7.22971406716846235638615219682, −6.86446202868112225942005641972, −5.67146648383425150727128106771, −4.42576871113998487592556013873, −3.53482430954986010512362618800, −3.04749952796369779989099980398, −0.75225984816335023561017403361, 1.40603675595129022331060214429, 3.29685514053679649567387590385, 3.82069517137348938609806377371, 4.77788934393264968171957370775, 6.00834262388701750858168146605, 6.76521959067287731198373268770, 7.69513893728417562561247170889, 8.493906475597883726163147533761, 9.201383350048747791169424894950, 9.802103852325838666151277210305

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