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Properties

Label 1-96-96.5-r1-0-0
Degree $1$
Conductor $96$
Sign $0.980 + 0.195i$
Analytic cond. $10.3166$
Root an. cond. $10.3166$
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  + (−0.707 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(96\)    =    \(2^{5} \cdot 3\)
Sign: $0.980 + 0.195i$
Analytic conductor: \(10.3166\)
Root analytic conductor: \(10.3166\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{96} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 96,\ (1:\ ),\ 0.980 + 0.195i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.611389837 + 0.1587080464i\)
\(L(\frac12)\) \(\approx\) \(1.611389837 + 0.1587080464i\)
\(L(1)\) \(\approx\) \(1.110332781 + 0.02935413730i\)
\(L(1)\) \(\approx\) \(1.110332781 + 0.02935413730i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 + iT \)
11 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + T \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 - iT \)
29 \( 1 + (0.707 - 0.707i)T \)
31 \( 1 + T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + T \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (-0.707 + 0.707i)T \)
67 \( 1 + (-0.707 + 0.707i)T \)
71 \( 1 - iT \)
73 \( 1 - iT \)
79 \( 1 - T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 - iT \)
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

−30.06780000530807022043397040875, −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