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Properties

Label 1-927-927.220-r0-0-0
Degree $1$
Conductor $927$
Sign $0.196 + 0.980i$
Analytic cond. $4.30496$
Root an. cond. $4.30496$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.196 + 0.980i$
Analytic conductor: \(4.30496\)
Root analytic conductor: \(4.30496\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (220, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (0:\ ),\ 0.196 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.631187994 + 2.157112210i\)
\(L(\frac12)\) \(\approx\) \(2.631187994 + 2.157112210i\)
\(L(1)\) \(\approx\) \(2.035673970 + 0.8964450494i\)
\(L(1)\) \(\approx\) \(2.035673970 + 0.8964450494i\)

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

−21.64761798671149585319800937374, −21.0093085190224014412029900746, −20.25170045086711999268841446154, −19.64260405276440640690190459308, −18.82052323280100107062155526776, −17.372435575242604456319670478921, −16.739859837896966420509137525139, −16.425771160885294156231288530145, −14.889481845860573367972201274132, −14.40838937004739867676877202775, −13.72852106281417537758275232113, −12.83180160266973967200875652202, −12.25285071428443873301204195337, −11.34349533047833178144889866313, −10.33621894285772258163925108148, −9.75188537684821845162437490162, −8.62924170750617721231971656184, −7.4245509186438603295075218296, −6.65331270747243337980231263146, −5.6825170022668802756231284413, −4.86189212176241669017100591938, −4.087906786439337308179755023872, −3.182288181799027595225642309167, −1.753893725457911302107623854234, −1.18520879888593414268300189137, 1.7337873639597485123131035255, 2.581436334205261875033914307186, 3.27934328083679440717980830974, 4.60610270040516697748466719139, 5.36984220417229995406522109554, 6.21072055355032750871790535956, 6.93373443483302147320243097454, 7.78670063434783630168719838181, 9.04088241465245030683147303743, 9.85376075465943869170887417708, 10.98365726413021461522830986942, 11.67030106105370636772467418261, 12.4895579321924864783431607158, 13.21316876793487597616298938752, 14.37269582879949298080945639812, 14.69879284756711514587731801137, 15.27870103519467282130676864868, 16.430112788154057365165007267616, 17.26896906913220308173711965180, 17.88280493409019902216171909626, 19.046581916941595151139024357053, 19.64560507177483897319502105722, 20.84218569624399063579155525220, 21.463884124323233896812399861583, 22.096975788529877225370335212997

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