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Properties

Label 1-189-189.38-r0-0-0
Degree $1$
Conductor $189$
Sign $0.276 + 0.960i$
Analytic cond. $0.877712$
Root an. cond. $0.877712$
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  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.939 + 0.342i)5-s + (0.5 + 0.866i)8-s − 10-s + (0.939 + 0.342i)11-s + (−0.173 + 0.984i)13-s + (0.173 + 0.984i)16-s + 17-s − 19-s + (−0.939 − 0.342i)20-s + (0.766 + 0.642i)22-s + (−0.173 + 0.984i)23-s + (0.766 − 0.642i)25-s + (−0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.939 + 0.342i)5-s + (0.5 + 0.866i)8-s − 10-s + (0.939 + 0.342i)11-s + (−0.173 + 0.984i)13-s + (0.173 + 0.984i)16-s + 17-s − 19-s + (−0.939 − 0.342i)20-s + (0.766 + 0.642i)22-s + (−0.173 + 0.984i)23-s + (0.766 − 0.642i)25-s + (−0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.276 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.276 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.276 + 0.960i$
Analytic conductor: \(0.877712\)
Root analytic conductor: \(0.877712\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (0:\ ),\ 0.276 + 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.413625679 + 1.063869410i\)
\(L(\frac12)\) \(\approx\) \(1.413625679 + 1.063869410i\)
\(L(1)\) \(\approx\) \(1.463157292 + 0.6086440336i\)
\(L(1)\) \(\approx\) \(1.463157292 + 0.6086440336i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.939 + 0.342i)T \)
5 \( 1 + (-0.939 + 0.342i)T \)
11 \( 1 + (0.939 + 0.342i)T \)
13 \( 1 + (-0.173 + 0.984i)T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (-0.173 - 0.984i)T \)
31 \( 1 + (-0.766 - 0.642i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (0.766 - 0.642i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (0.173 - 0.984i)T \)
61 \( 1 + (-0.766 + 0.642i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (0.173 + 0.984i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.766 + 0.642i)T \)
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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

−27.47102116258209289508399236670, −25.71338244577495549811824103356, −24.75047863230054791836899024541, −23.94788249634170836660976771375, −23.04449402773578715130486315207, −22.3221467319057146415921066809, −21.22536329197013012178558642780, −20.19656628800022766924188062262, −19.57691428460480133169571793084, −18.61180858020359528766853691844, −16.89433557971221017690488699590, −16.09438921097048842717530926967, −14.950789325945662820924489642685, −14.34172737719045004063746114493, −12.8316871943930149007619794720, −12.293306745820110940001083815252, −11.233960857490392017512127549577, −10.28270342452780097448373189232, −8.734487023029092407747958622016, −7.50493146155736476348472795536, −6.28820514387749530462745372837, −5.04861775366164574938964554613, −3.96360001648145335143306284641, −3.00596434074552736758314852285, −1.17245381742964331552035898931, 2.07449841684879524550400154454, 3.68699773546998143542685834758, 4.26073226809152623242075320751, 5.787167466782156375192162142766, 6.96339049202747388804044892500, 7.701532806585455829434667491801, 9.091446518052062563687478003795, 10.75068733515081890827120160711, 11.81075713264991378373239586262, 12.3301869943644189865914638567, 13.791921272855003424892696542080, 14.65208516955310153824902993879, 15.3959236479171538417662233052, 16.49716278882331891412607074051, 17.27566261748684632622008644385, 18.9579651852635933896495472936, 19.65858566492090957643901514656, 20.81978843035659339500142994913, 21.78484479305199793847153129523, 22.710725925424493526170515519147, 23.48383010618474666465254219484, 24.206559221608277368079106208985, 25.37008114738457459697471498689, 26.148330310015632865468206870227, 27.26753649042878700270309825155

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