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Properties

Label 1-84-84.11-r0-0-0
Degree $1$
Conductor $84$
Sign $0.832 - 0.553i$
Analytic cond. $0.390094$
Root an. cond. $0.390094$
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  + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)11-s + 13-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s − 29-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)37-s − 41-s − 43-s + (−0.5 + 0.866i)47-s + (0.5 + 0.866i)53-s − 55-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)11-s + 13-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s − 29-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)37-s − 41-s − 43-s + (−0.5 + 0.866i)47-s + (0.5 + 0.866i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.832 - 0.553i$
Analytic conductor: \(0.390094\)
Root analytic conductor: \(0.390094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 84,\ (0:\ ),\ 0.832 - 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9967769186 - 0.3012402018i\)
\(L(\frac12)\) \(\approx\) \(0.9967769186 - 0.3012402018i\)
\(L(1)\) \(\approx\) \(1.074967861 - 0.1837676773i\)
\(L(1)\) \(\approx\) \(1.074967861 - 0.1837676773i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (0.5 - 0.866i)T \)
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

−30.776101587770146938268129710393, −29.95026009123736014800726882183, −28.85471009034554244206355806159, −27.803961935896119077731209301355, −26.47489173174527488269798563393, −25.72911120558788813215907059910, −24.70474429028522404909683822056, −23.12427578416343378828026062659, −22.56515090861883152409931987495, −21.152274655726625266797544842406, −20.322027749755712132292159509471, −18.56681417231314503918701508207, −18.201901952651810769618017156568, −16.71169046993097783809968955934, −15.42726839865949797725185485791, −14.32265113883198674062414790567, −13.28592039162675572756403749578, −11.83691920232330516858510860407, −10.52936911897666048367824033756, −9.60860000931374621903166449591, −7.9111900103702069238136723198, −6.67307256621895801390059901298, −5.384314828234950571265997298464, −3.564680043885898841644550193032, −2.05185206543427806999833595911, 1.43379100056766816760335678131, 3.421243326092674281052487571783, 5.12199412191506097153837757958, 6.17895488106447636775583241984, 8.03934954937995374176232526471, 9.02124800570937866710137893216, 10.37608083676911194096067850761, 11.70602426091678641774294455588, 13.119632906927244427614493881, 13.799259918333861361662834776802, 15.50943266012893065845738897838, 16.47593214606831048913512484588, 17.56440889893560054951336434780, 18.74343109562507872501967674928, 20.03523669101452045643676811431, 21.09282213879159448654536614754, 21.88323939986568305355820530210, 23.51736342574846920645172794948, 24.217514882104440980528074946820, 25.44704753240674077354157787663, 26.33485601221573148492831030560, 27.793898664215444188694843991929, 28.54168642358002635848246799936, 29.57250083145304393826134011646, 30.70240701022664127605941315582

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