Properties

Label 1-927-927.331-r1-0-0
Degree $1$
Conductor $927$
Sign $0.773 - 0.633i$
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  + (0.969 − 0.243i)2-s + (0.881 − 0.473i)4-s + (−0.650 + 0.759i)5-s + (0.213 − 0.976i)7-s + (0.739 − 0.673i)8-s + (−0.445 + 0.895i)10-s + (−0.969 + 0.243i)11-s + (−0.952 + 0.303i)13-s + (−0.0307 − 0.999i)14-s + (0.552 − 0.833i)16-s + (0.932 + 0.361i)17-s + (−0.602 + 0.798i)19-s + (−0.213 + 0.976i)20-s + (−0.881 + 0.473i)22-s + (0.969 + 0.243i)23-s + ⋯
L(s)  = 1  + (0.969 − 0.243i)2-s + (0.881 − 0.473i)4-s + (−0.650 + 0.759i)5-s + (0.213 − 0.976i)7-s + (0.739 − 0.673i)8-s + (−0.445 + 0.895i)10-s + (−0.969 + 0.243i)11-s + (−0.952 + 0.303i)13-s + (−0.0307 − 0.999i)14-s + (0.552 − 0.833i)16-s + (0.932 + 0.361i)17-s + (−0.602 + 0.798i)19-s + (−0.213 + 0.976i)20-s + (−0.881 + 0.473i)22-s + (0.969 + 0.243i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.773 - 0.633i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ 0.773 - 0.633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.286536923 - 1.174745943i\)
\(L(\frac12)\) \(\approx\) \(3.286536923 - 1.174745943i\)
\(L(1)\) \(\approx\) \(1.732036099 - 0.2863578783i\)
\(L(1)\) \(\approx\) \(1.732036099 - 0.2863578783i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.969 - 0.243i)T \)
5 \( 1 + (-0.650 + 0.759i)T \)
7 \( 1 + (0.213 - 0.976i)T \)
11 \( 1 + (-0.969 + 0.243i)T \)
13 \( 1 + (-0.952 + 0.303i)T \)
17 \( 1 + (0.932 + 0.361i)T \)
19 \( 1 + (-0.602 + 0.798i)T \)
23 \( 1 + (0.969 + 0.243i)T \)
29 \( 1 + (0.332 + 0.943i)T \)
31 \( 1 + (0.998 + 0.0615i)T \)
37 \( 1 + (-0.0922 - 0.995i)T \)
41 \( 1 + (0.332 - 0.943i)T \)
43 \( 1 + (0.908 - 0.417i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.602 - 0.798i)T \)
59 \( 1 + (0.213 + 0.976i)T \)
61 \( 1 + (-0.153 - 0.988i)T \)
67 \( 1 + (-0.213 - 0.976i)T \)
71 \( 1 + (0.982 - 0.183i)T \)
73 \( 1 + (0.982 - 0.183i)T \)
79 \( 1 + (0.332 + 0.943i)T \)
83 \( 1 + (0.213 - 0.976i)T \)
89 \( 1 + (0.850 - 0.526i)T \)
97 \( 1 + (-0.779 - 0.626i)T \)
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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

−21.61602742823196901581494277133, −21.13722931349574676530633544811, −20.45719373851951967983766651221, −19.43912226095372984802469254972, −18.84956977380307374897595285537, −17.547280811384906528228176724028, −16.80412409029385406998896266815, −15.954299292917013037671475036381, −15.26837236409290628023564986679, −14.85154710489656229673286545800, −13.59745340254871889999750016177, −12.88496383495336622343570423273, −12.17515369673353586468273399590, −11.64405220041399080617210964675, −10.65310469045419677605836458737, −9.44288429626505696761774271247, −8.266236300422683527585932814770, −7.87528241228710171725129651444, −6.78578671448999434696137811849, −5.590586033248528070724052623801, −5.042534515680676413900691640928, −4.36041125349372871463829263409, −2.94812166056404465577442936534, −2.46723096290513317433821241407, −0.84122793321392802191702010270, 0.671819921206010215680165742850, 2.04898331156839703133666484459, 3.01587876898109088058639896463, 3.84946418999251393667274479341, 4.64976780144715335325743465524, 5.570275403450705375595592738580, 6.77341227828025326260468436498, 7.3791547505031329760851071413, 8.02332666127357158360375239869, 9.762900481927432088120723788245, 10.67386393992395302966742621022, 10.83214071940966253290396213238, 12.17241894163053884268148886561, 12.55287529029234823256635533631, 13.726060082203042181732195948012, 14.40419419361055570873950199064, 14.94614647346766287199374748957, 15.83142898401303263842477910523, 16.644063892416257694520683671408, 17.54279975216074514939766190951, 18.80763518037126062688473111741, 19.30355398601575984684435122042, 20.05986291781333503084046199668, 21.05348623528202187066330365044, 21.3934414798312423547074347611

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