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Properties

Label 1-927-927.176-r0-0-0
Degree $1$
Conductor $927$
Sign $0.777 + 0.628i$
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  + (0.908 + 0.417i)2-s + (0.650 + 0.759i)4-s + (−0.998 + 0.0615i)5-s + (−0.696 − 0.717i)7-s + (0.273 + 0.961i)8-s + (−0.932 − 0.361i)10-s + (−0.908 − 0.417i)11-s + (0.969 − 0.243i)13-s + (−0.332 − 0.943i)14-s + (−0.153 + 0.988i)16-s + (0.602 + 0.798i)17-s + (0.739 − 0.673i)19-s + (−0.696 − 0.717i)20-s + (−0.650 − 0.759i)22-s + (0.908 − 0.417i)23-s + ⋯
L(s)  = 1  + (0.908 + 0.417i)2-s + (0.650 + 0.759i)4-s + (−0.998 + 0.0615i)5-s + (−0.696 − 0.717i)7-s + (0.273 + 0.961i)8-s + (−0.932 − 0.361i)10-s + (−0.908 − 0.417i)11-s + (0.969 − 0.243i)13-s + (−0.332 − 0.943i)14-s + (−0.153 + 0.988i)16-s + (0.602 + 0.798i)17-s + (0.739 − 0.673i)19-s + (−0.696 − 0.717i)20-s + (−0.650 − 0.759i)22-s + (0.908 − 0.417i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.777 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.777 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.777 + 0.628i$
Analytic conductor: \(4.30496\)
Root analytic conductor: \(4.30496\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (0:\ ),\ 0.777 + 0.628i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.931083702 + 0.6833471796i\)
\(L(\frac12)\) \(\approx\) \(1.931083702 + 0.6833471796i\)
\(L(1)\) \(\approx\) \(1.454817825 + 0.3582616591i\)
\(L(1)\) \(\approx\) \(1.454817825 + 0.3582616591i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.908 + 0.417i)T \)
5 \( 1 + (-0.998 + 0.0615i)T \)
7 \( 1 + (-0.696 - 0.717i)T \)
11 \( 1 + (-0.908 - 0.417i)T \)
13 \( 1 + (0.969 - 0.243i)T \)
17 \( 1 + (0.602 + 0.798i)T \)
19 \( 1 + (0.739 - 0.673i)T \)
23 \( 1 + (0.908 - 0.417i)T \)
29 \( 1 + (-0.552 - 0.833i)T \)
31 \( 1 + (0.779 + 0.626i)T \)
37 \( 1 + (0.850 + 0.526i)T \)
41 \( 1 + (-0.552 + 0.833i)T \)
43 \( 1 + (0.0307 + 0.999i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.739 - 0.673i)T \)
59 \( 1 + (0.696 - 0.717i)T \)
61 \( 1 + (0.992 - 0.122i)T \)
67 \( 1 + (0.696 - 0.717i)T \)
71 \( 1 + (0.445 + 0.895i)T \)
73 \( 1 + (-0.445 - 0.895i)T \)
79 \( 1 + (0.552 + 0.833i)T \)
83 \( 1 + (0.696 + 0.717i)T \)
89 \( 1 + (-0.982 - 0.183i)T \)
97 \( 1 + (-0.389 - 0.920i)T \)
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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

−21.86179762527851971896126170518, −20.7978869921921297086488725706, −20.51133434445619086461880866340, −19.47471633775881548077388186007, −18.67505229682629866137955589473, −18.38049369153307924475406114581, −16.54914896940669517252163098328, −15.980302537737614974755305365401, −15.40855744420974887222865485822, −14.68938255180143071457841364585, −13.55426997860058978222135014039, −12.928046357909540674217548613375, −12.06675800559528719676309851404, −11.58306255667876092824304889331, −10.62540418008785673147827092508, −9.733467353376440576687493304932, −8.7477242803871688922721880980, −7.55987151148481225332628570473, −6.86812760779150977811554213228, −5.64721691652467528496562884485, −5.12005269518915527552102746014, −3.864862123326475214653443359526, −3.263049956155854519752856544422, −2.34570776919794058757681168175, −0.90662944073819583920818251518, 0.95374939027979934572267498114, 2.88122362263377973642073167726, 3.36496458282897460249712457804, 4.24910309383250136891298222452, 5.17525770223566707789212503731, 6.24469455826749292487373327786, 6.962259806794633450612206950114, 7.94556903213426056028623021100, 8.362336417293428944462117023252, 9.90401509063178459247215231104, 11.00002896787291972969130160895, 11.37434041401203125377352539012, 12.64136292722822707481418354951, 13.10364267472306232501280975552, 13.84247778241047192611391896648, 14.91887890863463616627109760378, 15.575217932830412476195394959485, 16.23686978409963337261584334561, 16.776367798674805674588638216956, 17.92749147236126986386183761410, 18.985065348395968430104184313314, 19.68262372293377275125287450717, 20.56927777707396117798548035411, 21.10431562223117574820900337956, 22.189843458453771497975781370402

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