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Properties

Label 1-1176-1176.125-r0-0-0
Degree $1$
Conductor $1176$
Sign $0.987 + 0.159i$
Analytic cond. $5.46132$
Root an. cond. $5.46132$
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  + (0.900 + 0.433i)5-s + (0.623 − 0.781i)11-s + (0.623 − 0.781i)13-s + (−0.222 + 0.974i)17-s + 19-s + (0.222 + 0.974i)23-s + (0.623 + 0.781i)25-s + (−0.222 + 0.974i)29-s − 31-s + (0.222 − 0.974i)37-s + (−0.900 − 0.433i)41-s + (0.900 − 0.433i)43-s + (0.623 − 0.781i)47-s + (−0.222 − 0.974i)53-s + (0.900 − 0.433i)55-s + ⋯
L(s)  = 1  + (0.900 + 0.433i)5-s + (0.623 − 0.781i)11-s + (0.623 − 0.781i)13-s + (−0.222 + 0.974i)17-s + 19-s + (0.222 + 0.974i)23-s + (0.623 + 0.781i)25-s + (−0.222 + 0.974i)29-s − 31-s + (0.222 − 0.974i)37-s + (−0.900 − 0.433i)41-s + (0.900 − 0.433i)43-s + (0.623 − 0.781i)47-s + (−0.222 − 0.974i)53-s + (0.900 − 0.433i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.987 + 0.159i$
Analytic conductor: \(5.46132\)
Root analytic conductor: \(5.46132\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1176,\ (0:\ ),\ 0.987 + 0.159i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.019547107 + 0.1621993060i\)
\(L(\frac12)\) \(\approx\) \(2.019547107 + 0.1621993060i\)
\(L(1)\) \(\approx\) \(1.363049983 + 0.06094392414i\)
\(L(1)\) \(\approx\) \(1.363049983 + 0.06094392414i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.900 + 0.433i)T \)
11 \( 1 + (0.623 - 0.781i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
17 \( 1 + (-0.222 + 0.974i)T \)
19 \( 1 + T \)
23 \( 1 + (0.222 + 0.974i)T \)
29 \( 1 + (-0.222 + 0.974i)T \)
31 \( 1 - T \)
37 \( 1 + (0.222 - 0.974i)T \)
41 \( 1 + (-0.900 - 0.433i)T \)
43 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (0.623 - 0.781i)T \)
53 \( 1 + (-0.222 - 0.974i)T \)
59 \( 1 + (0.900 - 0.433i)T \)
61 \( 1 + (-0.222 + 0.974i)T \)
67 \( 1 - T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (-0.623 - 0.781i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.623 - 0.781i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
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

−20.99540700323005959413962819977, −20.57440492270316110655570710595, −19.8991415298289745918259240554, −18.685065765262518288473412329835, −18.1931573439321831785502128998, −17.32825272853041027748570801641, −16.63997142235359327979706534027, −15.96964157423330013326148276018, −14.92924927290955448816401722771, −14.04765177733213355310606782677, −13.58012229642618859393582047879, −12.63117589366801802395263997144, −11.85267542688735038005614049903, −11.04796836507559914013249116543, −9.90921261542516065263902156443, −9.37119652397429198458425513740, −8.73091779777266864133281592239, −7.51659947207365944284886646464, −6.66076444048277811728912294970, −5.91990243982627322943582490229, −4.87520566426132115892388968030, −4.22954783539306369038877577117, −2.90967870738815000289645297318, −1.92699533122942462009424068116, −1.051239469617448875906040803230, 1.09092793829899297422504063394, 1.982699747085784780676313413560, 3.25344349982372023624967045155, 3.75322768645958034487784129483, 5.42189374646590762905754262452, 5.71981870850606307983727670137, 6.72737603302936975760650894579, 7.57082763334127946767832529911, 8.73209473075604950450600993957, 9.26166396397636616913017123226, 10.35219459067727358048717544662, 10.88025940780965726959704052808, 11.74331523272096329942764324777, 12.89598534602745671046429444051, 13.452449734369789377844423444034, 14.250739541704462502208115620215, 14.926936746756104313589919483871, 15.8883549918483123361787694771, 16.69020507245266301826367160296, 17.54967937161983547407133624215, 18.062127850297125891057787880360, 18.90000858069822423269532163236, 19.72009552402307325249735518327, 20.526206934168087455299253147618, 21.37321582611486742335790193210

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