Properties

Label 1-927-927.641-r1-0-0
Degree $1$
Conductor $927$
Sign $0.388 + 0.921i$
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  + (−0.992 − 0.122i)2-s + (0.969 + 0.243i)4-s + (0.908 + 0.417i)5-s + (−0.779 − 0.626i)7-s + (−0.932 − 0.361i)8-s + (−0.850 − 0.526i)10-s + (−0.992 − 0.122i)11-s + (−0.153 + 0.988i)13-s + (0.696 + 0.717i)14-s + (0.881 + 0.473i)16-s + (0.982 − 0.183i)17-s + (0.445 − 0.895i)19-s + (0.779 + 0.626i)20-s + (0.969 + 0.243i)22-s + (−0.992 + 0.122i)23-s + ⋯
L(s)  = 1  + (−0.992 − 0.122i)2-s + (0.969 + 0.243i)4-s + (0.908 + 0.417i)5-s + (−0.779 − 0.626i)7-s + (−0.932 − 0.361i)8-s + (−0.850 − 0.526i)10-s + (−0.992 − 0.122i)11-s + (−0.153 + 0.988i)13-s + (0.696 + 0.717i)14-s + (0.881 + 0.473i)16-s + (0.982 − 0.183i)17-s + (0.445 − 0.895i)19-s + (0.779 + 0.626i)20-s + (0.969 + 0.243i)22-s + (−0.992 + 0.122i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.388 + 0.921i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ 0.388 + 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7771784180 + 0.5156387721i\)
\(L(\frac12)\) \(\approx\) \(0.7771784180 + 0.5156387721i\)
\(L(1)\) \(\approx\) \(0.6944423353 + 0.02237468275i\)
\(L(1)\) \(\approx\) \(0.6944423353 + 0.02237468275i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.992 - 0.122i)T \)
5 \( 1 + (0.908 + 0.417i)T \)
7 \( 1 + (-0.779 - 0.626i)T \)
11 \( 1 + (-0.992 - 0.122i)T \)
13 \( 1 + (-0.153 + 0.988i)T \)
17 \( 1 + (0.982 - 0.183i)T \)
19 \( 1 + (0.445 - 0.895i)T \)
23 \( 1 + (-0.992 + 0.122i)T \)
29 \( 1 + (-0.816 + 0.577i)T \)
31 \( 1 + (-0.0307 - 0.999i)T \)
37 \( 1 + (0.739 - 0.673i)T \)
41 \( 1 + (-0.816 - 0.577i)T \)
43 \( 1 + (0.213 - 0.976i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.445 + 0.895i)T \)
59 \( 1 + (0.779 - 0.626i)T \)
61 \( 1 + (0.650 + 0.759i)T \)
67 \( 1 + (-0.779 + 0.626i)T \)
71 \( 1 + (-0.0922 + 0.995i)T \)
73 \( 1 + (0.0922 - 0.995i)T \)
79 \( 1 + (0.816 - 0.577i)T \)
83 \( 1 + (0.779 + 0.626i)T \)
89 \( 1 + (0.273 - 0.961i)T \)
97 \( 1 + (0.332 + 0.943i)T \)
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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

−21.29743392780014388082783269892, −20.61887718352605731498111190972, −19.95890876822627189301538277925, −18.9293542116601260622166243322, −18.25519848972588376514523594331, −17.77925940683765722474419050389, −16.65433144676226732136805188789, −16.26021099872083237121115278827, −15.35132689009243367083317379773, −14.55200022184719060602361735525, −13.34839094408237536919654112104, −12.56849063479220944197393808441, −11.91953752941220253723849610650, −10.52214672999967921187839673174, −9.95768937062412115199180841934, −9.51585462570457217428838107922, −8.293040183286420082763670242798, −7.86110739572250294946359104658, −6.55779979403096860561324426802, −5.717019783160882503289016992653, −5.29166640085510779870663904534, −3.335312165332479245778304140830, −2.528085989146105585254904440737, −1.55587118921100931529143673745, −0.34694440543101119245608662198, 0.800435594865947795622233136637, 2.04601404723238521058312961965, 2.80598574830077932499785720753, 3.81523848943127643166970099035, 5.42549683567450757091256216710, 6.21827503657655683726356485836, 7.16272723245438602238349364563, 7.665376340569680661194778995729, 9.060333580256345635550539447231, 9.61520595577808745819098994843, 10.305561389806169204146586431402, 10.98380245812986289410462723157, 11.99062945916946438717698447489, 13.02350699354263104053872941799, 13.74540810569890094427024994466, 14.66330355818808449671789887112, 15.79605718778325254741825491342, 16.420511000509643988085583535202, 17.11537691281561364856108116693, 17.92017232334167185099889123755, 18.72324674772048999366704683361, 19.14679129521519146796062995684, 20.31550907415515176605276065160, 20.77519800546813486007635243541, 21.76442244333285400132342077523

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