Properties

Label 1-927-927.130-r1-0-0
Degree $1$
Conductor $927$
Sign $-0.329 - 0.944i$
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  + (0.332 − 0.943i)2-s + (−0.779 − 0.626i)4-s + (0.389 − 0.920i)5-s + (0.881 − 0.473i)7-s + (−0.850 + 0.526i)8-s + (−0.739 − 0.673i)10-s + (−0.332 + 0.943i)11-s + (−0.0307 + 0.999i)13-s + (−0.153 − 0.988i)14-s + (0.213 + 0.976i)16-s + (−0.273 + 0.961i)17-s + (0.0922 − 0.995i)19-s + (−0.881 + 0.473i)20-s + (0.779 + 0.626i)22-s + (0.332 + 0.943i)23-s + ⋯
L(s)  = 1  + (0.332 − 0.943i)2-s + (−0.779 − 0.626i)4-s + (0.389 − 0.920i)5-s + (0.881 − 0.473i)7-s + (−0.850 + 0.526i)8-s + (−0.739 − 0.673i)10-s + (−0.332 + 0.943i)11-s + (−0.0307 + 0.999i)13-s + (−0.153 − 0.988i)14-s + (0.213 + 0.976i)16-s + (−0.273 + 0.961i)17-s + (0.0922 − 0.995i)19-s + (−0.881 + 0.473i)20-s + (0.779 + 0.626i)22-s + (0.332 + 0.943i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.329 - 0.944i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ -0.329 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.587718775 - 2.236160421i\)
\(L(\frac12)\) \(\approx\) \(1.587718775 - 2.236160421i\)
\(L(1)\) \(\approx\) \(1.114699854 - 0.8237259474i\)
\(L(1)\) \(\approx\) \(1.114699854 - 0.8237259474i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.332 - 0.943i)T \)
5 \( 1 + (0.389 - 0.920i)T \)
7 \( 1 + (0.881 - 0.473i)T \)
11 \( 1 + (-0.332 + 0.943i)T \)
13 \( 1 + (-0.0307 + 0.999i)T \)
17 \( 1 + (-0.273 + 0.961i)T \)
19 \( 1 + (0.0922 - 0.995i)T \)
23 \( 1 + (0.332 + 0.943i)T \)
29 \( 1 + (0.992 - 0.122i)T \)
31 \( 1 + (0.952 + 0.303i)T \)
37 \( 1 + (-0.445 - 0.895i)T \)
41 \( 1 + (0.992 + 0.122i)T \)
43 \( 1 + (-0.552 - 0.833i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.0922 + 0.995i)T \)
59 \( 1 + (0.881 + 0.473i)T \)
61 \( 1 + (-0.696 - 0.717i)T \)
67 \( 1 + (-0.881 - 0.473i)T \)
71 \( 1 + (0.602 - 0.798i)T \)
73 \( 1 + (0.602 - 0.798i)T \)
79 \( 1 + (0.992 - 0.122i)T \)
83 \( 1 + (0.881 - 0.473i)T \)
89 \( 1 + (-0.932 - 0.361i)T \)
97 \( 1 + (0.969 + 0.243i)T \)
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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

−22.22711613907235330189304015036, −21.13074093648441230782236885266, −20.822506244358307915236877638170, −19.20337763974571186314353391595, −18.39670056789423163134449796994, −17.99223806553774290878059693992, −17.21786963609131498462147139854, −16.1828897035426263547418897109, −15.471227228602666985878719581699, −14.677981445596066423425612191862, −14.1043456382417398372327131257, −13.422099701558980946028840767668, −12.370374245525073367203408237811, −11.44953795241689436345919199205, −10.54051739935987553848174245854, −9.62083996012781917957063790809, −8.386515700091838911188120771392, −8.02689851501281517924297757010, −6.94019991263043917511323642125, −6.06489844340203308401398306020, −5.41613454081087504433153764016, −4.51847407599022795488274659148, −3.16854302044054956128373936265, −2.590277325083900746994144485368, −0.82174665809460699957785552084, 0.71316454108044308061146920723, 1.67677987786472924012457194251, 2.28649372177199936391689141809, 3.83981725070063855675525551757, 4.66480001854601937057029271118, 5.05223170068418726873158245301, 6.2847165373233878757430513483, 7.51702741302990435890731870698, 8.632390677096715076853147556191, 9.229427737245033434636279389520, 10.17539218701747285914506563597, 10.91335317065073810015479313066, 11.88886215817068168785050423017, 12.43823521706131398374691985321, 13.50527880171552528014070753298, 13.82054718343146989221315244466, 14.910064850386377011109530647076, 15.684716578163527499074226030025, 17.06298908068700877641994546618, 17.51998047233573241293274814884, 18.17872045630142734055823972970, 19.529292341292176232829793341328, 19.80901255262881039148003432308, 20.85032489571977472433717090474, 21.25552798005947902948561075175

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