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Properties

Label 1-927-927.236-r1-0-0
Degree $1$
Conductor $927$
Sign $0.979 + 0.200i$
Analytic cond. $99.6199$
Root an. cond. $99.6199$
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  + (−0.816 + 0.577i)2-s + (0.332 − 0.943i)4-s + (−0.552 − 0.833i)5-s + (0.969 − 0.243i)7-s + (0.273 + 0.961i)8-s + (0.932 + 0.361i)10-s + (−0.816 + 0.577i)11-s + (−0.696 − 0.717i)13-s + (−0.650 + 0.759i)14-s + (−0.779 − 0.626i)16-s + (0.602 + 0.798i)17-s + (0.739 − 0.673i)19-s + (−0.969 + 0.243i)20-s + (0.332 − 0.943i)22-s + (−0.816 − 0.577i)23-s + ⋯
L(s)  = 1  + (−0.816 + 0.577i)2-s + (0.332 − 0.943i)4-s + (−0.552 − 0.833i)5-s + (0.969 − 0.243i)7-s + (0.273 + 0.961i)8-s + (0.932 + 0.361i)10-s + (−0.816 + 0.577i)11-s + (−0.696 − 0.717i)13-s + (−0.650 + 0.759i)14-s + (−0.779 − 0.626i)16-s + (0.602 + 0.798i)17-s + (0.739 − 0.673i)19-s + (−0.969 + 0.243i)20-s + (0.332 − 0.943i)22-s + (−0.816 − 0.577i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.979 + 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.979 + 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.979 + 0.200i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ 0.979 + 0.200i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.132486703 + 0.1146623839i\)
\(L(\frac12)\) \(\approx\) \(1.132486703 + 0.1146623839i\)
\(L(1)\) \(\approx\) \(0.7145513908 + 0.04593634933i\)
\(L(1)\) \(\approx\) \(0.7145513908 + 0.04593634933i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.816 + 0.577i)T \)
5 \( 1 + (-0.552 - 0.833i)T \)
7 \( 1 + (0.969 - 0.243i)T \)
11 \( 1 + (-0.816 + 0.577i)T \)
13 \( 1 + (-0.696 - 0.717i)T \)
17 \( 1 + (0.602 + 0.798i)T \)
19 \( 1 + (0.739 - 0.673i)T \)
23 \( 1 + (-0.816 - 0.577i)T \)
29 \( 1 + (0.998 - 0.0615i)T \)
31 \( 1 + (-0.153 + 0.988i)T \)
37 \( 1 + (-0.850 - 0.526i)T \)
41 \( 1 + (0.998 + 0.0615i)T \)
43 \( 1 + (0.881 + 0.473i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.739 + 0.673i)T \)
59 \( 1 + (-0.969 - 0.243i)T \)
61 \( 1 + (-0.389 + 0.920i)T \)
67 \( 1 + (0.969 + 0.243i)T \)
71 \( 1 + (-0.445 - 0.895i)T \)
73 \( 1 + (0.445 + 0.895i)T \)
79 \( 1 + (-0.998 + 0.0615i)T \)
83 \( 1 + (-0.969 + 0.243i)T \)
89 \( 1 + (0.982 + 0.183i)T \)
97 \( 1 + (0.992 + 0.122i)T \)
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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

−21.51376069669422350783498839598, −20.802619569721341340407928230112, −20.02086247570785550682029414000, −19.006284377392778986788991013156, −18.6108090770088173452723112489, −17.956935546992091677563143627045, −17.08919535178393603703460307369, −16.0623308523261972446485288817, −15.54790906962590895577648382518, −14.30630149034795548343847734511, −13.82183064253215588971177748509, −12.3145536614410057267845371980, −11.76981879983246749004419327119, −11.16763542360182850687952504873, −10.30398269041862918579761138454, −9.57003120716931701001526604646, −8.42056494013093925807661211555, −7.70392585709816566105176953286, −7.25089591444578501384298576172, −5.90909989843032677185430442749, −4.707156374431222773064738371777, −3.61435269674668579116126014424, −2.72708718172763348115718245142, −1.891322839186307110412593238846, −0.544401897773292995100122413333, 0.624928637565493979827192357109, 1.50626918520399639825129353585, 2.706359706011759184379745907364, 4.39512626803507097703946617944, 5.030189948748822649057587508663, 5.7990356767151661350649561875, 7.26430046111085688602250960042, 7.788168211151994408922533060587, 8.354586710681137803672317943975, 9.31345294308850846737820154256, 10.31057194480157935491654223068, 10.88982423958787338391313946389, 12.05359977706503219785573598738, 12.661491766757485254688658756708, 13.99457366111067817666509686144, 14.661392980336369166059600319853, 15.656254278741096115868194653705, 15.97987569000057480404042544515, 17.182547805177335251680740301709, 17.57207805373203698191988078064, 18.301932105099038696490556388836, 19.42480073162264539695638771838, 20.00597139181688627596697286325, 20.63564789318721654756559745710, 21.441154107549057505151114105729

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