Properties

Degree 1
Conductor $ 2^{5} \cdot 3 $
Sign $0.980 + 0.195i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.707 − 0.707i)5-s + i·7-s + (0.707 + 0.707i)11-s + (0.707 − 0.707i)13-s + 17-s + (0.707 − 0.707i)19-s i·23-s + i·25-s + (0.707 − 0.707i)29-s + 31-s + (0.707 − 0.707i)35-s + (0.707 + 0.707i)37-s i·41-s + (−0.707 − 0.707i)43-s + 47-s + ⋯
L(s,χ)  = 1  + (−0.707 − 0.707i)5-s + i·7-s + (0.707 + 0.707i)11-s + (0.707 − 0.707i)13-s + 17-s + (0.707 − 0.707i)19-s i·23-s + i·25-s + (0.707 − 0.707i)29-s + 31-s + (0.707 − 0.707i)35-s + (0.707 + 0.707i)37-s i·41-s + (−0.707 − 0.707i)43-s + 47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.980 + 0.195i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 96 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.980 + 0.195i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(96\)    =    \(2^{5} \cdot 3\)
\( \varepsilon \)  =  $0.980 + 0.195i$
motivic weight  =  \(0\)
character  :  $\chi_{96} (5, \cdot )$
Sato-Tate  :  $\mu(8)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 96,\ (1:\ ),\ 0.980 + 0.195i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.611389837 + 0.1587080464i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.611389837 + 0.1587080464i\)
\(L(\chi,1)\)  \(\approx\)  \(1.110332781 + 0.02935413730i\)
\(L(1,\chi)\)  \(\approx\)  \(1.110332781 + 0.02935413730i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−30.06780000530807022043397040875, −28.93586760342209735418563732538, −27.51413994339351579896532783860, −26.79306585245525887578514059094, −25.93987067365356100806488372936, −24.523929503885352899949104252531, −23.38733824280815638060295893766, −22.75812226839694104739836272267, −21.436192535522458616280110976295, −20.23066252331653853008805495826, −19.19691462213747725701357501708, −18.33645195054829484632969122950, −16.783609494395250160594774670217, −16.04928243523343412894953958928, −14.446549905838767790462846682469, −13.89152305195750078878177030631, −12.15029855861848490925566273925, −11.1295159653169572579630183034, −10.12241776940605967675900800351, −8.477952631020275895854840584467, −7.28868641414182450036389336344, −6.210802523091507033331748949991, −4.2152145947679326418778188120, −3.2646141563764406285246752740, −0.98964930777336152307311400613, 1.166732917775604093400979712308, 3.15524006242734913070103360686, 4.67108455333818432791040048523, 5.91242826953153538874492881068, 7.604627286508713238559046217948, 8.690188329504745449121054272981, 9.78901838130667050699121018088, 11.604100373293786385403437687171, 12.20835156412595002355839005209, 13.48614510296297047281258762987, 15.121695344159545248999059962649, 15.732253205665777662176177635298, 17.04812575886473079834097781307, 18.222444561504455679133454896389, 19.410647240091604699417345995168, 20.32863478739375870972951677628, 21.44144370204961519363072164647, 22.68459638671549212819460758706, 23.59709678160239991436010703208, 24.87656433421228531279907004133, 25.4519306173124855223075160356, 27.062615852906699731787547363860, 28.022128508890453750454777304048, 28.475027779121691759809983891909, 30.132994679943752617937604493408

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