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Properties

Label 1-1132-1132.1131-r0-0-0
Degree $1$
Conductor $1132$
Sign $1$
Analytic cond. $5.25698$
Root an. cond. $5.25698$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 7-s + 9-s − 11-s + 13-s − 15-s − 17-s + 19-s − 21-s − 23-s + 25-s + 27-s + 29-s + 31-s − 33-s + 35-s − 37-s + 39-s + 41-s + 43-s − 45-s + 47-s + 49-s − 51-s − 53-s + 55-s + ⋯
L(s)  = 1  + 3-s − 5-s − 7-s + 9-s − 11-s + 13-s − 15-s − 17-s + 19-s − 21-s − 23-s + 25-s + 27-s + 29-s + 31-s − 33-s + 35-s − 37-s + 39-s + 41-s + 43-s − 45-s + 47-s + 49-s − 51-s − 53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1132 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1132 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1132\)    =    \(2^{2} \cdot 283\)
Sign: $1$
Analytic conductor: \(5.25698\)
Root analytic conductor: \(5.25698\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1132} (1131, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1132,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.540528178\)
\(L(\frac12)\) \(\approx\) \(1.540528178\)
\(L(1)\) \(\approx\) \(1.155462955\)
\(L(1)\) \(\approx\) \(1.155462955\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
283 \( 1 \)
good3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 - T \)
19 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 + T \)
31 \( 1 + T \)
37 \( 1 - T \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 - T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 + T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.10598889372569795326063608762, −20.38975334141527801898801611406, −19.83711738017582028588880086682, −19.1016860388706359114637763410, −18.52521625334742003785855268116, −17.68871109921855861732812478240, −16.12023320353729699417655601421, −15.73152220866103473699271377746, −15.554809872248463945272973551997, −14.141936171065846490281989555765, −13.59182655096039912530571630479, −12.77517811546835958468772067188, −12.08941185804149295298204320432, −10.92476640429911759216709171941, −10.2024451759444396224560995908, −9.25099405288814290540619941517, −8.47215007726050387721947336691, −7.80587281782518085782488597014, −6.995192434637311778612364540450, −6.084709574800303173844926626228, −4.70629837557771193784137595749, −3.86093941991935579209550541814, −3.14372118232533964394987917531, −2.3693945378061778078438175524, −0.821063992085654819976009985676, 0.821063992085654819976009985676, 2.3693945378061778078438175524, 3.14372118232533964394987917531, 3.86093941991935579209550541814, 4.70629837557771193784137595749, 6.084709574800303173844926626228, 6.995192434637311778612364540450, 7.80587281782518085782488597014, 8.47215007726050387721947336691, 9.25099405288814290540619941517, 10.2024451759444396224560995908, 10.92476640429911759216709171941, 12.08941185804149295298204320432, 12.77517811546835958468772067188, 13.59182655096039912530571630479, 14.141936171065846490281989555765, 15.554809872248463945272973551997, 15.73152220866103473699271377746, 16.12023320353729699417655601421, 17.68871109921855861732812478240, 18.52521625334742003785855268116, 19.1016860388706359114637763410, 19.83711738017582028588880086682, 20.38975334141527801898801611406, 21.10598889372569795326063608762

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