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Properties

Label 1-927-927.301-r1-0-0
Degree $1$
Conductor $927$
Sign $-0.950 - 0.310i$
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  + (−0.153 + 0.988i)2-s + (−0.952 − 0.303i)4-s + (−0.969 + 0.243i)5-s + (−0.998 − 0.0615i)7-s + (0.445 − 0.895i)8-s + (−0.0922 − 0.995i)10-s + (0.153 − 0.988i)11-s + (0.552 − 0.833i)13-s + (0.213 − 0.976i)14-s + (0.816 + 0.577i)16-s + (−0.850 − 0.526i)17-s + (−0.982 − 0.183i)19-s + (0.998 + 0.0615i)20-s + (0.952 + 0.303i)22-s + (−0.153 − 0.988i)23-s + ⋯
L(s)  = 1  + (−0.153 + 0.988i)2-s + (−0.952 − 0.303i)4-s + (−0.969 + 0.243i)5-s + (−0.998 − 0.0615i)7-s + (0.445 − 0.895i)8-s + (−0.0922 − 0.995i)10-s + (0.153 − 0.988i)11-s + (0.552 − 0.833i)13-s + (0.213 − 0.976i)14-s + (0.816 + 0.577i)16-s + (−0.850 − 0.526i)17-s + (−0.982 − 0.183i)19-s + (0.998 + 0.0615i)20-s + (0.952 + 0.303i)22-s + (−0.153 − 0.988i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.950 - 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.950 - 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.950 - 0.310i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ -0.950 - 0.310i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05358916240 - 0.3369650341i\)
\(L(\frac12)\) \(\approx\) \(0.05358916240 - 0.3369650341i\)
\(L(1)\) \(\approx\) \(0.5766456365 + 0.07803409835i\)
\(L(1)\) \(\approx\) \(0.5766456365 + 0.07803409835i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.153 + 0.988i)T \)
5 \( 1 + (-0.969 + 0.243i)T \)
7 \( 1 + (-0.998 - 0.0615i)T \)
11 \( 1 + (0.153 - 0.988i)T \)
13 \( 1 + (0.552 - 0.833i)T \)
17 \( 1 + (-0.850 - 0.526i)T \)
19 \( 1 + (-0.982 - 0.183i)T \)
23 \( 1 + (-0.153 - 0.988i)T \)
29 \( 1 + (-0.696 - 0.717i)T \)
31 \( 1 + (0.908 - 0.417i)T \)
37 \( 1 + (0.602 - 0.798i)T \)
41 \( 1 + (-0.696 + 0.717i)T \)
43 \( 1 + (-0.992 + 0.122i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.982 + 0.183i)T \)
59 \( 1 + (-0.998 + 0.0615i)T \)
61 \( 1 + (0.881 - 0.473i)T \)
67 \( 1 + (0.998 - 0.0615i)T \)
71 \( 1 + (0.273 + 0.961i)T \)
73 \( 1 + (0.273 + 0.961i)T \)
79 \( 1 + (-0.696 - 0.717i)T \)
83 \( 1 + (-0.998 - 0.0615i)T \)
89 \( 1 + (-0.739 - 0.673i)T \)
97 \( 1 + (-0.0307 - 0.999i)T \)
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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

−21.99533317167821318755616444353, −21.1481142588223967674433814625, −20.19398809302914419098397386007, −19.731967238044568813861596681069, −19.057686624252875407359345526922, −18.40464139395307376408905697964, −17.28499734966712246539542781649, −16.64097396088541822476108939811, −15.60971142030291776993206625658, −14.94569722535212165528053170911, −13.701705989892231713955979248339, −12.955686017746760605236818610044, −12.30993687455937303184609849341, −11.60096918281758852647305308838, −10.76154667726358073642350131926, −9.871372326923201190055534641455, −9.02293686513131124713829577656, −8.404697460568137719161678325120, −7.2554680295472432540046126541, −6.38045304683370709759994796482, −4.93003835351347926112555475794, −4.05832320456268606864764303209, −3.548724234293045695374168200881, −2.302387337569462401805924463008, −1.29256360972584977585589152526, 0.143531727939260805542768060951, 0.602822723226014082309907140034, 2.74032885154571838341140789790, 3.7377852091426535047377176449, 4.42650944501418928004378510465, 5.74713572180413405403037307907, 6.46599705775534416921832811298, 7.13456706625102836728356407153, 8.32493554146837107096133562703, 8.597295859858452454122177277017, 9.800243918863977750277604592858, 10.6694870464147060389319057083, 11.53466587704929796603651605581, 12.81551935242851643845925653879, 13.28388859958647690673416090758, 14.25356062902899458919692566859, 15.33690881047128372931120088858, 15.59931654852117676560115712139, 16.505968355887526374512628211183, 17.03713176927108668152097224126, 18.36611821026929580340865095332, 18.717234881697262666846303204455, 19.5950607536384546647119616146, 20.21319303247716556991072121787, 21.64249781542455087434576603696

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