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Properties

Label 1-72e2-5184.211-r1-0-0
Degree $1$
Conductor $5184$
Sign $0.994 - 0.107i$
Analytic cond. $557.098$
Root an. cond. $557.098$
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  + (0.912 + 0.409i)5-s + (−0.620 + 0.784i)7-s + (−0.775 + 0.631i)11-s + (−0.328 + 0.944i)13-s + (0.342 − 0.939i)17-s + (0.999 + 0.0436i)19-s + (0.620 + 0.784i)23-s + (0.664 + 0.747i)25-s + (−0.0726 − 0.997i)29-s + (0.597 − 0.802i)31-s + (−0.887 + 0.461i)35-s + (0.461 − 0.887i)37-s + (−0.979 − 0.202i)41-s + (0.159 + 0.987i)43-s + (0.802 − 0.597i)47-s + ⋯
L(s)  = 1  + (0.912 + 0.409i)5-s + (−0.620 + 0.784i)7-s + (−0.775 + 0.631i)11-s + (−0.328 + 0.944i)13-s + (0.342 − 0.939i)17-s + (0.999 + 0.0436i)19-s + (0.620 + 0.784i)23-s + (0.664 + 0.747i)25-s + (−0.0726 − 0.997i)29-s + (0.597 − 0.802i)31-s + (−0.887 + 0.461i)35-s + (0.461 − 0.887i)37-s + (−0.979 − 0.202i)41-s + (0.159 + 0.987i)43-s + (0.802 − 0.597i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $0.994 - 0.107i$
Analytic conductor: \(557.098\)
Root analytic conductor: \(557.098\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5184} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5184,\ (1:\ ),\ 0.994 - 0.107i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.440412916 - 0.1317527274i\)
\(L(\frac12)\) \(\approx\) \(2.440412916 - 0.1317527274i\)
\(L(1)\) \(\approx\) \(1.168144841 + 0.1941142635i\)
\(L(1)\) \(\approx\) \(1.168144841 + 0.1941142635i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.912 + 0.409i)T \)
7 \( 1 + (-0.620 + 0.784i)T \)
11 \( 1 + (-0.775 + 0.631i)T \)
13 \( 1 + (-0.328 + 0.944i)T \)
17 \( 1 + (0.342 - 0.939i)T \)
19 \( 1 + (0.999 + 0.0436i)T \)
23 \( 1 + (0.620 + 0.784i)T \)
29 \( 1 + (-0.0726 - 0.997i)T \)
31 \( 1 + (0.597 - 0.802i)T \)
37 \( 1 + (0.461 - 0.887i)T \)
41 \( 1 + (-0.979 - 0.202i)T \)
43 \( 1 + (0.159 + 0.987i)T \)
47 \( 1 + (0.802 - 0.597i)T \)
53 \( 1 + (0.608 - 0.793i)T \)
59 \( 1 + (0.101 + 0.994i)T \)
61 \( 1 + (0.696 + 0.717i)T \)
67 \( 1 + (0.756 - 0.653i)T \)
71 \( 1 + (0.819 + 0.573i)T \)
73 \( 1 + (-0.819 + 0.573i)T \)
79 \( 1 + (0.549 - 0.835i)T \)
83 \( 1 + (0.561 + 0.827i)T \)
89 \( 1 + (-0.573 - 0.819i)T \)
97 \( 1 + (0.686 - 0.727i)T \)
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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

−17.67879175748318176981235412462, −17.15081632265831664583163378442, −16.59260789412289529455290155615, −15.96444589719664221659025158704, −15.252930009686983170332374729756, −14.31370775790049634984904694623, −13.80131037428024700782216498859, −13.08646952703269720514909053497, −12.75071055908443689930613459171, −11.988328343456798122148067776357, −10.68588621049040513484986178898, −10.51307938976326845669110448799, −9.87783677440106471460245901267, −9.07265826904049698142955872560, −8.32051111952829085463257480252, −7.6870074889105882615959971749, −6.757537114134007712488433254, −6.19156779613535959148301438228, −5.24338780466549770437799239659, −5.01091530723815834116985762621, −3.734524775064104269134253753315, −3.08437925478729999335292467035, −2.41769066823657751609710191526, −1.13951851562030355112682769984, −0.7842416227483036788134990427, 0.41664532505794288250864469117, 1.57839223643621306198870505714, 2.49617275659689283118895684383, 2.719328115513221518459074827604, 3.78164607101622432206351639140, 4.85627658714282883708350525859, 5.4800066427518970506766375532, 5.995317693269446736683206846452, 7.03077539107979509384486781174, 7.2782518879760955749661853802, 8.371837219095976254109181640550, 9.38222951014666165580178708563, 9.62648233038736373420399980351, 10.109684233654747609018802786298, 11.23083019029767947074957616726, 11.76795620140066026701736441558, 12.457032316419583311088926763383, 13.41923639767584024770493132162, 13.570666996426207843669147837369, 14.54099408689509383386147797493, 15.145263168117098093731488974913, 15.807382898100304561115959864103, 16.46960340705328799371893269779, 17.18650986112464788313245241362, 17.94948891737178234232680343319

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