Properties

Label 1-231-231.164-r1-0-0
Degree $1$
Conductor $231$
Sign $-0.991 - 0.126i$
Analytic cond. $24.8243$
Root an. cond. $24.8243$
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)2-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 8-s + (−0.5 − 0.866i)10-s + 13-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + 20-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (−0.5 + 0.866i)26-s + 29-s + (0.5 + 0.866i)31-s + (−0.5 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 8-s + (−0.5 − 0.866i)10-s + 13-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + 20-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (−0.5 + 0.866i)26-s + 29-s + (0.5 + 0.866i)31-s + (−0.5 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.991 - 0.126i$
Analytic conductor: \(24.8243\)
Root analytic conductor: \(24.8243\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 231,\ (1:\ ),\ -0.991 - 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05041065982 + 0.7943667364i\)
\(L(\frac12)\) \(\approx\) \(-0.05041065982 + 0.7943667364i\)
\(L(1)\) \(\approx\) \(0.5698728510 + 0.4335339153i\)
\(L(1)\) \(\approx\) \(0.5698728510 + 0.4335339153i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
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

−25.734636040260262799338858544103, −24.88606156588350940139027231086, −23.54243368868539408595151892697, −22.89405144008357451423745864785, −21.52038367360644036365152502937, −20.879765413465973143166516906756, −19.99751203894546182112525002111, −19.229810131655245009393151554647, −18.23584832972740245658269244438, −17.23287138442905329814136931241, −16.34116414317045073487735348017, −15.42834701273704681722804939169, −13.69882756273328963991086454971, −13.04394091703409142431042541698, −11.89052183517308165399467667546, −11.28387024666155855651680084871, −10.00913656163928561753211633137, −8.94796085210545034438977438089, −8.26481040085242567256053244351, −7.073822524868902224155675476292, −5.247711728013957351743954985558, −4.17214479442646865606330026096, −3.0570613065144894930051298196, −1.46853525338644026977366003605, −0.34543172885145348334654443492, 1.411384482692903596729552619856, 3.29636121454598608162825425472, 4.563507476474801986456470594373, 6.09227477959860989454018215602, 6.7166808322927764782960754758, 8.002919225277005233801725263837, 8.628959237859456398669364121638, 10.22927047001386459046282827335, 10.7018436860016602406917062547, 12.117389606329563488808608491279, 13.54492948449322864301721944742, 14.5102178013077563132438892356, 15.23913412750183803104140281827, 16.157309136498564600785869396596, 17.112217459750719319261440081722, 18.241083679253083005820156173033, 18.85177216846684172249995424379, 19.670139322038965228332559247055, 21.00991231006536251219090822034, 22.26835570067120376505542436388, 23.23346238808895929710590996955, 23.61792860054613623186147306227, 24.984987574968944513719296177813, 25.6978077769602727483003421773, 26.53353235390391998324999606471

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