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Properties

Label 1-864-864.11-r0-0-0
Degree $1$
Conductor $864$
Sign $-0.614 + 0.788i$
Analytic cond. $4.01239$
Root an. cond. $4.01239$
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  + (0.0871 + 0.996i)5-s + (−0.342 + 0.939i)7-s + (0.996 + 0.0871i)11-s + (0.573 + 0.819i)13-s + (−0.5 + 0.866i)17-s + (−0.965 − 0.258i)19-s + (0.342 + 0.939i)23-s + (−0.984 + 0.173i)25-s + (−0.819 − 0.573i)29-s + (0.939 − 0.342i)31-s + (−0.965 − 0.258i)35-s + (0.965 − 0.258i)37-s + (0.984 + 0.173i)41-s + (−0.996 − 0.0871i)43-s + (0.939 + 0.342i)47-s + ⋯
L(s)  = 1  + (0.0871 + 0.996i)5-s + (−0.342 + 0.939i)7-s + (0.996 + 0.0871i)11-s + (0.573 + 0.819i)13-s + (−0.5 + 0.866i)17-s + (−0.965 − 0.258i)19-s + (0.342 + 0.939i)23-s + (−0.984 + 0.173i)25-s + (−0.819 − 0.573i)29-s + (0.939 − 0.342i)31-s + (−0.965 − 0.258i)35-s + (0.965 − 0.258i)37-s + (0.984 + 0.173i)41-s + (−0.996 − 0.0871i)43-s + (0.939 + 0.342i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.614 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.614 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.614 + 0.788i$
Analytic conductor: \(4.01239\)
Root analytic conductor: \(4.01239\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 864,\ (0:\ ),\ -0.614 + 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5579483135 + 1.141855286i\)
\(L(\frac12)\) \(\approx\) \(0.5579483135 + 1.141855286i\)
\(L(1)\) \(\approx\) \(0.9373678842 + 0.4581992759i\)
\(L(1)\) \(\approx\) \(0.9373678842 + 0.4581992759i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.0871 + 0.996i)T \)
7 \( 1 + (-0.342 + 0.939i)T \)
11 \( 1 + (0.996 + 0.0871i)T \)
13 \( 1 + (0.573 + 0.819i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.965 - 0.258i)T \)
23 \( 1 + (0.342 + 0.939i)T \)
29 \( 1 + (-0.819 - 0.573i)T \)
31 \( 1 + (0.939 - 0.342i)T \)
37 \( 1 + (0.965 - 0.258i)T \)
41 \( 1 + (0.984 + 0.173i)T \)
43 \( 1 + (-0.996 - 0.0871i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (-0.0871 - 0.996i)T \)
61 \( 1 + (-0.906 + 0.422i)T \)
67 \( 1 + (0.573 + 0.819i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (-0.819 - 0.573i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (0.766 - 0.642i)T \)
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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

−21.75179229630744393149287590698, −20.72483213858824794213136491429, −20.23614661945815930825240390943, −19.6477854284199599088606308893, −18.616857507538708440575168942177, −17.58741419568753317644059625401, −16.8795563135117277984790119987, −16.393173868815800245651840052784, −15.46237724695802804971827245433, −14.43324774274412095636007992144, −13.54893887559624234315995784568, −12.95207960079101777548572636129, −12.15467478553070516829560755659, −11.1085544996404715107965408575, −10.32161455003817509840952810052, −9.311246870742299518926529249395, −8.66689376896000728663595316863, −7.7168694427211422323659601003, −6.66978274688176224170128792403, −5.9009412924059107548390504526, −4.64429097729209794632783840849, −4.09387737803715310740728853984, −2.9490937306044185460707473736, −1.48529033685646054532417592234, −0.585243578355683022349855683989, 1.66591464871504534656900478497, 2.485573788046610424918471904704, 3.5990732416073213882846096626, 4.38146832978653835530339463764, 6.01156834522056707486375992656, 6.2595486708912801658990377055, 7.220870449715626961469380674453, 8.430080324619911115903856460630, 9.21998337407814537958782657960, 9.94861154614280076356806841347, 11.23901535510376776627774626398, 11.458185944113860424188886945142, 12.65056859136640663788925551332, 13.478493810390696716085656409832, 14.44480548728955660606991049323, 15.10441189679320863273277981878, 15.70425255509533044879338652559, 16.89004125067095310798918300464, 17.558854301831628731873458797606, 18.50683020607591560247786143467, 19.1992143674406636679845345014, 19.58354848867496783616931287830, 20.997189288172573773691082170704, 21.71981579045043188351942205577, 22.1633424879314381421600323695

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