Properties

Degree $1$
Conductor $111$
Sign $-0.989 - 0.146i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s − 8-s + 10-s − 11-s + (−0.5 + 0.866i)13-s − 14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 + 0.866i)20-s + (−0.5 − 0.866i)22-s − 23-s + ⋯
L(s,χ)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s − 8-s + 10-s − 11-s + (−0.5 + 0.866i)13-s − 14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 + 0.866i)20-s + (−0.5 − 0.866i)22-s − 23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.989 - 0.146i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.989 - 0.146i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $-0.989 - 0.146i$
Motivic weight: \(0\)
Character: $\chi_{111} (47, \cdot )$
Sato-Tate group: $\mu(6)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 111,\ (1:\ ),\ -0.989 - 0.146i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(-0.07887491130 + 1.070448415i\)
\(L(\frac12,\chi)\) \(\approx\) \(-0.07887491130 + 1.070448415i\)
\(L(\chi,1)\) \(\approx\) \(0.8091982405 + 0.6420143822i\)
\(L(1,\chi)\) \(\approx\) \(0.8091982405 + 0.6420143822i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−29.04011021407130301433877348060, −27.81029902145922628456788369425, −26.64542845891587597688897537011, −25.842247904423617382866291949025, −24.3180959651082133028021820355, −23.10253290695865750618921415046, −22.54982634279637854154781713158, −21.44646093799440661683677016668, −20.44687132591694384248374350924, −19.45537130779909021567985925925, −18.38212512045739802358073849025, −17.48200876554257249189164512825, −15.76125249364560042070237171885, −14.581267262568288079842685496711, −13.579764963398494281280847834550, −12.817854122647055173999895033238, −11.27248306765815981696051685978, −10.31068609808432651578502367190, −9.67811649802846502695256963625, −7.62731440951895220857773280234, −6.241073119025989824259167768168, −4.95079886168676955216569141280, −3.35940341473298417511494220270, −2.402553993857765072666991137804, −0.3591970261784863868965153983, 2.29970471625148554418487555071, 4.09818938062467910082188359130, 5.4375427904857280217309907488, 6.16469324950065084056711957540, 7.84070898076470212140781127074, 8.84304095015318157918840941672, 9.94766723615973758378241989334, 12.12334595741048527651872230903, 12.72573990131036388752816957709, 13.83594668107369036175094266545, 15.039832706034207057474743600111, 16.127211460970822633056731562766, 16.84985451708250434797902867394, 18.04456375785926877968317863854, 19.17553124797370789454543544909, 20.95320419650824854668885595119, 21.47747357255398036992091689190, 22.61645411431743078110287578518, 23.872441642366467895301386234395, 24.49259829039940053932478281789, 25.63426022890338312760043433046, 26.19077178659394999543793535578, 27.74175945838338535918619834918, 28.66492597248760423002848457194, 29.72768021350767095459682906893

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