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Properties

Label 2-132-12.11-c1-0-8
Degree $2$
Conductor $132$
Sign $0.667 + 0.744i$
Analytic cond. $1.05402$
Root an. cond. $1.02665$
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  + (−0.621 + 1.27i)2-s + (−1.72 + 0.121i)3-s + (−1.22 − 1.57i)4-s − 2.24i·5-s + (0.919 − 2.27i)6-s − 2i·7-s + (2.76 − 0.578i)8-s + (2.97 − 0.419i)9-s + (2.84 + 1.39i)10-s + 11-s + (2.31 + 2.57i)12-s − 5.08·13-s + (2.54 + 1.24i)14-s + (0.272 + 3.87i)15-s + (−0.985 + 3.87i)16-s − 6.91i·17-s + ⋯
L(s)  = 1  + (−0.439 + 0.898i)2-s + (−0.997 + 0.0700i)3-s + (−0.613 − 0.789i)4-s − 1.00i·5-s + (0.375 − 0.926i)6-s − 0.755i·7-s + (0.978 − 0.204i)8-s + (0.990 − 0.139i)9-s + (0.900 + 0.440i)10-s + 0.301·11-s + (0.667 + 0.744i)12-s − 1.40·13-s + (0.679 + 0.332i)14-s + (0.0702 + 1.00i)15-s + (−0.246 + 0.969i)16-s − 1.67i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(132\)    =    \(2^{2} \cdot 3 \cdot 11\)
Sign: $0.667 + 0.744i$
Analytic conductor: \(1.05402\)
Root analytic conductor: \(1.02665\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{132} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 132,\ (\ :1/2),\ 0.667 + 0.744i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.477116 - 0.212974i\)
\(L(\frac12)\) \(\approx\) \(0.477116 - 0.212974i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.621 - 1.27i)T \)
3 \( 1 + (1.72 - 0.121i)T \)
11 \( 1 - T \)
good5 \( 1 + 2.24iT - 5T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
13 \( 1 + 5.08T + 13T^{2} \)
17 \( 1 + 6.91iT - 17T^{2} \)
19 \( 1 + 5.08iT - 19T^{2} \)
23 \( 1 - 0.859T + 23T^{2} \)
29 \( 1 - 4iT - 29T^{2} \)
31 \( 1 - 4.24iT - 31T^{2} \)
37 \( 1 - 0.222T + 37T^{2} \)
41 \( 1 - 3.39iT - 41T^{2} \)
43 \( 1 - 1.82iT - 43T^{2} \)
47 \( 1 + 1.40T + 47T^{2} \)
53 \( 1 - 6.59iT - 53T^{2} \)
59 \( 1 - 10.7T + 59T^{2} \)
61 \( 1 + 2.31T + 61T^{2} \)
67 \( 1 + 14.5iT - 67T^{2} \)
71 \( 1 - 7.02T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 + 10.5iT - 79T^{2} \)
83 \( 1 + 9.07T + 83T^{2} \)
89 \( 1 + 2.67iT - 89T^{2} \)
97 \( 1 + 3.02T + 97T^{2} \)
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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

−13.18241240314507409772014738252, −12.17030363089210808207760809444, −10.97454241226956449367919186444, −9.802632444266438655620081127034, −9.067961334940694458961087643212, −7.44932107878270642600550952104, −6.78701310275213628489098090070, −5.08906012046710500255675852973, −4.70774721804159287816256988926, −0.72937455160458739825495610244, 2.19003778657254760035728928516, 3.97678942186486620033630707180, 5.61017328481896379303047971566, 6.93566481144213862819255202946, 8.163943350617431261330025122024, 9.752956108550352491540444371209, 10.38760015945557794034410916711, 11.37613443237475191690268131099, 12.19301679542524664211739250010, 12.86461926921513898807933005829

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