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Properties

Label 1-927-927.785-r1-0-0
Degree $1$
Conductor $927$
Sign $0.816 + 0.577i$
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  + (0.952 + 0.303i)2-s + (0.816 + 0.577i)4-s + (−0.881 + 0.473i)5-s + (0.992 + 0.122i)7-s + (0.602 + 0.798i)8-s + (−0.982 + 0.183i)10-s + (0.952 + 0.303i)11-s + (−0.389 − 0.920i)13-s + (0.908 + 0.417i)14-s + (0.332 + 0.943i)16-s + (−0.445 − 0.895i)17-s + (0.932 + 0.361i)19-s + (−0.992 − 0.122i)20-s + (0.816 + 0.577i)22-s + (0.952 − 0.303i)23-s + ⋯
L(s)  = 1  + (0.952 + 0.303i)2-s + (0.816 + 0.577i)4-s + (−0.881 + 0.473i)5-s + (0.992 + 0.122i)7-s + (0.602 + 0.798i)8-s + (−0.982 + 0.183i)10-s + (0.952 + 0.303i)11-s + (−0.389 − 0.920i)13-s + (0.908 + 0.417i)14-s + (0.332 + 0.943i)16-s + (−0.445 − 0.895i)17-s + (0.932 + 0.361i)19-s + (−0.992 − 0.122i)20-s + (0.816 + 0.577i)22-s + (0.952 − 0.303i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (785, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ 0.816 + 0.577i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.433786657 + 1.408323350i\)
\(L(\frac12)\) \(\approx\) \(4.433786657 + 1.408323350i\)
\(L(1)\) \(\approx\) \(2.033265399 + 0.5189658721i\)
\(L(1)\) \(\approx\) \(2.033265399 + 0.5189658721i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.952 + 0.303i)T \)
5 \( 1 + (-0.881 + 0.473i)T \)
7 \( 1 + (0.992 + 0.122i)T \)
11 \( 1 + (0.952 + 0.303i)T \)
13 \( 1 + (-0.389 - 0.920i)T \)
17 \( 1 + (-0.445 - 0.895i)T \)
19 \( 1 + (0.932 + 0.361i)T \)
23 \( 1 + (0.952 - 0.303i)T \)
29 \( 1 + (0.0307 - 0.999i)T \)
31 \( 1 + (0.650 - 0.759i)T \)
37 \( 1 + (-0.273 - 0.961i)T \)
41 \( 1 + (0.0307 + 0.999i)T \)
43 \( 1 + (0.969 - 0.243i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.932 - 0.361i)T \)
59 \( 1 + (-0.992 + 0.122i)T \)
61 \( 1 + (0.552 - 0.833i)T \)
67 \( 1 + (0.992 - 0.122i)T \)
71 \( 1 + (0.850 - 0.526i)T \)
73 \( 1 + (-0.850 + 0.526i)T \)
79 \( 1 + (-0.0307 + 0.999i)T \)
83 \( 1 + (-0.992 - 0.122i)T \)
89 \( 1 + (-0.0922 - 0.995i)T \)
97 \( 1 + (-0.998 + 0.0615i)T \)
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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

−21.670238918124147268553511230993, −20.825963008755217832684673300678, −20.109210305092699201189507052671, −19.44767703974282614092980300970, −18.81674306746024271122286163526, −17.40976839426848183910025886597, −16.74841679843384268748611055163, −15.81151906421545549411505736522, −15.09341747297037592058881277204, −14.317912703109322947154197873848, −13.71714347792196655708151374962, −12.58012081161780728612779099535, −11.898952907442292961410102496598, −11.34361157785309205170132495676, −10.653299863373398871073548778620, −9.26678435458537254732457593975, −8.49717642619200403871861928835, −7.32321479434981031885792883864, −6.739994625645951871735038302293, −5.42054478124930043439656865787, −4.65483787343096938856717658263, −4.031205783213622901473829321061, −3.09208367139639171949422602575, −1.67237120424435531222537141133, −1.00967707833996787624685830536, 0.881536852268281560686830871393, 2.33552037244588163377510719351, 3.1715405979484928880346205410, 4.23000043299857953659461943915, 4.825070442563853889799203767137, 5.84893090797435157369001521303, 6.9293738964745307708011761546, 7.60461490316486173597635072296, 8.22173832027318364184424014648, 9.502328580999742751399901903826, 10.84449348519749366784822112592, 11.43821712852419660170365337581, 12.02437903050591358225143549439, 12.85309688614784082290290063018, 14.11106381583779058023343823517, 14.44158299103636791251297222881, 15.38095375902907682919294704443, 15.751658101897332265666798935416, 16.98166401128025831967856903822, 17.592694774593365905803132110008, 18.59280468827276505206768755258, 19.63953660815044655990512339188, 20.33685320666114756798702718431, 20.9182685856607501262786493836, 22.01634681857537824559658635695

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