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Properties

Label 1-927-927.137-r1-0-0
Degree $1$
Conductor $927$
Sign $0.984 + 0.177i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.984 + 0.177i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ 0.984 + 0.177i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.857928894 + 0.1659956326i\)
\(L(\frac12)\) \(\approx\) \(1.857928894 + 0.1659956326i\)
\(L(1)\) \(\approx\) \(0.9754926645 + 0.2604681718i\)
\(L(1)\) \(\approx\) \(0.9754926645 + 0.2604681718i\)

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

−21.33108822654887590580936384062, −21.15343925809392525155353809670, −20.04261492904922676676759362351, −19.2390409135609800379851538225, −18.51822733527126162522955145216, −17.75580159330113880785188293403, −17.05639728905930242099915274037, −16.56449073374615220031900787234, −15.23342276543333827240608868008, −14.09801596710417116175934698785, −13.50055339830622056639792562646, −12.99822398779230427252815316294, −11.555029114637386331701433107447, −11.11434661878301153649424703287, −10.28471331093267523911251006590, −9.64952558777529753289975245177, −8.558861825737896441561107087044, −7.9787276383277207627849115147, −6.81091091435243978449791786324, −5.90222457570526171299421008976, −4.593054827156652487586755184, −3.82554431127561368374097716124, −2.6194417796786758098373424299, −1.80101091539710416976698071953, −0.89182427515030019893966463305, 0.58279316373284044544395476376, 1.805637211929359116703102154, 2.549857702287432896792724755044, 4.46871886042153720337626119396, 5.23342252016192173136445725745, 5.82535817301261704597336520577, 6.779577740182487233501215174835, 7.84214074874029948444477069110, 8.557962646315218727421206307560, 9.28127032137265486087884782790, 10.22662837455994281016707163410, 10.74389066461495077704861705827, 12.18601556654713874693748904271, 12.95469732320307355018104885999, 13.91681807984129956541527242764, 14.63932976033385627656404949571, 15.380431025397800864044264314721, 16.13537684075025079584133943018, 17.05526728714882741759380448842, 17.90900806588265154007591515042, 18.124077702344374589628498357066, 18.95929377745176983110347563538, 20.3769014072692056348151429258, 20.61692959639665652035336647942, 21.76739141665843293218435312559

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