Properties

Label 1-927-927.610-r1-0-0
Degree $1$
Conductor $927$
Sign $0.999 - 0.0336i$
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  + (−0.779 − 0.626i)2-s + (0.213 + 0.976i)4-s + (0.696 + 0.717i)5-s + (0.552 − 0.833i)7-s + (0.445 − 0.895i)8-s + (−0.0922 − 0.995i)10-s + (0.779 + 0.626i)11-s + (−0.998 − 0.0615i)13-s + (−0.952 + 0.303i)14-s + (−0.908 + 0.417i)16-s + (−0.850 − 0.526i)17-s + (−0.982 − 0.183i)19-s + (−0.552 + 0.833i)20-s + (−0.213 − 0.976i)22-s + (−0.779 + 0.626i)23-s + ⋯
L(s)  = 1  + (−0.779 − 0.626i)2-s + (0.213 + 0.976i)4-s + (0.696 + 0.717i)5-s + (0.552 − 0.833i)7-s + (0.445 − 0.895i)8-s + (−0.0922 − 0.995i)10-s + (0.779 + 0.626i)11-s + (−0.998 − 0.0615i)13-s + (−0.952 + 0.303i)14-s + (−0.908 + 0.417i)16-s + (−0.850 − 0.526i)17-s + (−0.982 − 0.183i)19-s + (−0.552 + 0.833i)20-s + (−0.213 − 0.976i)22-s + (−0.779 + 0.626i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.999 - 0.0336i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (610, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ 0.999 - 0.0336i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.555220477 + 0.02615960306i\)
\(L(\frac12)\) \(\approx\) \(1.555220477 + 0.02615960306i\)
\(L(1)\) \(\approx\) \(0.8717277568 - 0.1227832547i\)
\(L(1)\) \(\approx\) \(0.8717277568 - 0.1227832547i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.779 - 0.626i)T \)
5 \( 1 + (0.696 + 0.717i)T \)
7 \( 1 + (0.552 - 0.833i)T \)
11 \( 1 + (0.779 + 0.626i)T \)
13 \( 1 + (-0.998 - 0.0615i)T \)
17 \( 1 + (-0.850 - 0.526i)T \)
19 \( 1 + (-0.982 - 0.183i)T \)
23 \( 1 + (-0.779 + 0.626i)T \)
29 \( 1 + (0.969 - 0.243i)T \)
31 \( 1 + (-0.816 - 0.577i)T \)
37 \( 1 + (0.602 - 0.798i)T \)
41 \( 1 + (0.969 + 0.243i)T \)
43 \( 1 + (0.389 - 0.920i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.982 + 0.183i)T \)
59 \( 1 + (0.552 + 0.833i)T \)
61 \( 1 + (-0.0307 + 0.999i)T \)
67 \( 1 + (-0.552 - 0.833i)T \)
71 \( 1 + (0.273 + 0.961i)T \)
73 \( 1 + (0.273 + 0.961i)T \)
79 \( 1 + (0.969 - 0.243i)T \)
83 \( 1 + (0.552 - 0.833i)T \)
89 \( 1 + (-0.739 - 0.673i)T \)
97 \( 1 + (0.881 + 0.473i)T \)
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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

−21.73109787168661004177638314287, −20.845511992586872406934430369756, −19.783285456826492710006319470136, −19.379832829727335605235691603848, −18.21151668687103485880937950077, −17.71868766515180167975569770011, −16.90665576123984241327088467278, −16.387889155767653158376326560981, −15.35465668554783251960044198904, −14.555569200791553741883007289247, −14.015772778004211624468376178058, −12.77665645915860222347938914123, −11.97669376968796909067864496083, −10.970901108049905548648021983642, −10.08747400178401333274277673865, −9.14414002012578842291449562940, −8.66311712749426098508458804115, −7.97955082079705167503949253005, −6.58489274504034370628249982037, −6.0733683095328217054258430462, −5.115603879240597297849247609962, −4.35260640065859640059847678741, −2.392548120841833936321834679835, −1.78579222207376216519792615458, −0.58989350458511217587118243739, 0.740411760487409706391576967578, 2.00347215626170982819592767898, 2.46636374098908687639813168091, 3.89375396343439441019045739290, 4.5595062245943541424614966305, 6.10452713299605142731882534261, 7.18111223008552868450909414641, 7.45458984489207022455741867205, 8.809963895421612329506242713948, 9.596920515691329100097588116919, 10.26674944919314528501161982433, 10.99929998370360117428801426193, 11.73743246067668969725906365945, 12.71416289406897201460463980475, 13.63119501262779728838870332074, 14.3976863946783714616394261568, 15.2321055823886703019926180742, 16.45835302378751672637653371612, 17.418966519166846069097410654042, 17.53984023201399127308657074676, 18.36329809265039314753873097406, 19.57045126536279922647630254785, 19.82264011423839062586981492655, 20.81699268771488860212452656027, 21.580139119864833647473670813626

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