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Properties

Label 1-927-927.724-r1-0-0
Degree $1$
Conductor $927$
Sign $0.985 - 0.171i$
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  + (0.552 + 0.833i)2-s + (−0.389 + 0.920i)4-s + (0.952 + 0.303i)5-s + (0.650 − 0.759i)7-s + (−0.982 + 0.183i)8-s + (0.273 + 0.961i)10-s + (−0.552 − 0.833i)11-s + (0.332 + 0.943i)13-s + (0.992 + 0.122i)14-s + (−0.696 − 0.717i)16-s + (0.0922 − 0.995i)17-s + (−0.850 − 0.526i)19-s + (−0.650 + 0.759i)20-s + (0.389 − 0.920i)22-s + (0.552 − 0.833i)23-s + ⋯
L(s)  = 1  + (0.552 + 0.833i)2-s + (−0.389 + 0.920i)4-s + (0.952 + 0.303i)5-s + (0.650 − 0.759i)7-s + (−0.982 + 0.183i)8-s + (0.273 + 0.961i)10-s + (−0.552 − 0.833i)11-s + (0.332 + 0.943i)13-s + (0.992 + 0.122i)14-s + (−0.696 − 0.717i)16-s + (0.0922 − 0.995i)17-s + (−0.850 − 0.526i)19-s + (−0.650 + 0.759i)20-s + (0.389 − 0.920i)22-s + (0.552 − 0.833i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.985 - 0.171i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (724, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ 0.985 - 0.171i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.835011388 - 0.2444608513i\)
\(L(\frac12)\) \(\approx\) \(2.835011388 - 0.2444608513i\)
\(L(1)\) \(\approx\) \(1.489594445 + 0.4638915598i\)
\(L(1)\) \(\approx\) \(1.489594445 + 0.4638915598i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.552 + 0.833i)T \)
5 \( 1 + (0.952 + 0.303i)T \)
7 \( 1 + (0.650 - 0.759i)T \)
11 \( 1 + (-0.552 - 0.833i)T \)
13 \( 1 + (0.332 + 0.943i)T \)
17 \( 1 + (0.0922 - 0.995i)T \)
19 \( 1 + (-0.850 - 0.526i)T \)
23 \( 1 + (0.552 - 0.833i)T \)
29 \( 1 + (0.213 + 0.976i)T \)
31 \( 1 + (-0.969 + 0.243i)T \)
37 \( 1 + (-0.932 - 0.361i)T \)
41 \( 1 + (0.213 - 0.976i)T \)
43 \( 1 + (0.153 - 0.988i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.850 + 0.526i)T \)
59 \( 1 + (0.650 + 0.759i)T \)
61 \( 1 + (0.816 + 0.577i)T \)
67 \( 1 + (-0.650 - 0.759i)T \)
71 \( 1 + (-0.739 - 0.673i)T \)
73 \( 1 + (-0.739 - 0.673i)T \)
79 \( 1 + (0.213 + 0.976i)T \)
83 \( 1 + (0.650 - 0.759i)T \)
89 \( 1 + (0.602 - 0.798i)T \)
97 \( 1 + (-0.908 - 0.417i)T \)
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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

−21.543924336248746810076243363517, −20.90722688560557891995077755002, −20.50894612109454401291515269435, −19.37635868527464239102245556190, −18.608475453672755078909991979793, −17.67880617585201756287314460216, −17.43765014137705307230166535330, −15.886763546603599359298284165467, −14.91123685191295694925234813909, −14.64336476454767493451604771441, −13.31517527714111367707100072451, −12.919726351645005220082820605443, −12.20958268131376286241724029131, −11.14631945058168681728228785521, −10.373980490159797445442300225117, −9.71111336966386743601382254364, −8.760608946294918956045483284105, −7.931506445009103144199851237918, −6.341564580822131224884445147236, −5.600791899109324178478118905833, −5.03419230813249729792514517737, −3.99137505528676499315551084192, −2.724234194760751925549809922647, −1.97815623721570683564641376200, −1.20842751441117701279216011476, 0.48174897277130805491562679118, 2.00491242161395011948122376459, 3.051523165625919311361544158347, 4.1322624182403540644930621408, 5.0736265850468543466741385462, 5.73866266251943253828600034658, 6.94763840350454842272871551421, 7.15729523385451589978456054139, 8.68014846385169777626491798679, 8.95847988284286923898700339866, 10.46208899164305428077457631792, 11.019310550089698212994281089507, 12.14888849374284301288869094180, 13.25672875407829577958153338343, 13.77105878405641615715072512614, 14.29655266415691994903276883272, 15.09974780359852029446240412757, 16.33800658743720799891750607086, 16.655671467867361990626307538671, 17.610463162811105443339937720354, 18.23324444054571468736660591298, 19.00851502532975986217660576003, 20.51341472275737023312143423452, 21.056696855673244066488372050811, 21.65686789743277246779903725003

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