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Properties

Label 1-927-927.20-r0-0-0
Degree $1$
Conductor $927$
Sign $0.882 + 0.470i$
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  + (−0.932 − 0.361i)2-s + (0.739 + 0.673i)4-s + (−0.273 − 0.961i)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 + (0.552 + 0.833i)13-s + (−0.213 − 0.976i)14-s + (0.0922 + 0.995i)16-s + (−0.881 − 0.473i)17-s + (0.332 − 0.943i)19-s + (0.445 − 0.895i)20-s + (0.952 − 0.303i)22-s + (0.153 − 0.988i)23-s + ⋯
L(s)  = 1  + (−0.932 − 0.361i)2-s + (0.739 + 0.673i)4-s + (−0.273 − 0.961i)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 + (0.552 + 0.833i)13-s + (−0.213 − 0.976i)14-s + (0.0922 + 0.995i)16-s + (−0.881 − 0.473i)17-s + (0.332 − 0.943i)19-s + (0.445 − 0.895i)20-s + (0.952 − 0.303i)22-s + (0.153 − 0.988i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.882 + 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.882 + 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.882 + 0.470i$
Analytic conductor: \(4.30496\)
Root analytic conductor: \(4.30496\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (0:\ ),\ 0.882 + 0.470i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7802790713 + 0.1948514486i\)
\(L(\frac12)\) \(\approx\) \(0.7802790713 + 0.1948514486i\)
\(L(1)\) \(\approx\) \(0.6937016126 - 0.04251560907i\)
\(L(1)\) \(\approx\) \(0.6937016126 - 0.04251560907i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.932 - 0.361i)T \)
5 \( 1 + (-0.273 - 0.961i)T \)
7 \( 1 + (0.552 + 0.833i)T \)
11 \( 1 + (-0.779 + 0.626i)T \)
13 \( 1 + (0.552 + 0.833i)T \)
17 \( 1 + (-0.881 - 0.473i)T \)
19 \( 1 + (0.332 - 0.943i)T \)
23 \( 1 + (0.153 - 0.988i)T \)
29 \( 1 + (0.273 + 0.961i)T \)
31 \( 1 + (0.908 + 0.417i)T \)
37 \( 1 + (0.602 + 0.798i)T \)
41 \( 1 + (-0.969 + 0.243i)T \)
43 \( 1 + (0.602 - 0.798i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.650 + 0.759i)T \)
59 \( 1 + (-0.552 + 0.833i)T \)
61 \( 1 + (0.881 + 0.473i)T \)
67 \( 1 + (-0.445 - 0.895i)T \)
71 \( 1 + (0.969 - 0.243i)T \)
73 \( 1 + (0.273 - 0.961i)T \)
79 \( 1 + (-0.696 + 0.717i)T \)
83 \( 1 + (-0.445 + 0.895i)T \)
89 \( 1 + (0.739 - 0.673i)T \)
97 \( 1 + (-0.850 - 0.526i)T \)
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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

−21.6395348389888922202941627530, −20.83436162162287275223163227538, −20.06652263052442340241296623921, −19.27324919347109540710253534681, −18.56480673521792931285222896789, −17.79990910632703922847247068691, −17.331107521843892226657750703396, −16.186745120821071827221504477864, −15.547822025569966687736838856840, −14.85266681779088721259894213523, −13.948415666640057883711878069898, −13.207501620016933667562748038539, −11.614217538973262829478008012662, −11.10086489758544669829604437500, −10.39732719866333248656889004568, −9.81416125618834506706342074912, −8.2559097184433980102351592384, −8.02903507765649054449349746391, −7.12730288179557504375889983231, −6.20832556888424449296218129922, −5.41943320481056621326723917098, −3.97076602324116404947243085448, −2.97681761513692432673647999383, −1.87067388411965853196344001836, −0.57063209812890505640652382960, 1.02719548800904182528868907336, 2.0638563321894918638597503189, 2.88129459721483041699826617278, 4.41440868108061399699166177999, 5.00371909604485667588493633708, 6.375915934026710849107505890143, 7.29896637205057809627033653192, 8.32820343351423847888851442009, 8.81341542163002480327522579596, 9.46275259568298186689412438649, 10.61445948378017300336241629195, 11.45236716289559527975237421542, 12.08073827581751512991297561389, 12.82572114928731480023412852985, 13.7287547469569729090811013964, 15.20295300519500373802886329984, 15.70021273660694060126014454556, 16.412606733690085901581569594915, 17.3184855296457382330494480197, 18.126444262142391920347133707065, 18.576741957844108556252019685035, 19.64716725200855714813844965836, 20.31806850319143876611229544607, 20.96267056132886184175925975043, 21.55239233834527205661836921306

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