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Properties

Label 1-189-189.4-r0-0-0
Degree 11
Conductor 189189
Sign 0.983−0.178i0.983 - 0.178i
Analytic cond. 0.8777120.877712
Root an. cond. 0.8777120.877712
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)4-s + (0.766 − 0.642i)5-s + (−0.5 + 0.866i)8-s + (−0.5 + 0.866i)10-s + (0.766 + 0.642i)11-s + (0.766 − 0.642i)13-s + (0.173 − 0.984i)16-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.173 − 0.984i)20-s + (−0.939 − 0.342i)22-s + (−0.939 − 0.342i)23-s + (0.173 − 0.984i)25-s + (−0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)4-s + (0.766 − 0.642i)5-s + (−0.5 + 0.866i)8-s + (−0.5 + 0.866i)10-s + (0.766 + 0.642i)11-s + (0.766 − 0.642i)13-s + (0.173 − 0.984i)16-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.173 − 0.984i)20-s + (−0.939 − 0.342i)22-s + (−0.939 − 0.342i)23-s + (0.173 − 0.984i)25-s + (−0.5 + 0.866i)26-s + ⋯

Functional equation

Λ(s)=(189s/2 ΓR(s) L(s)=((0.983−0.178i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(189s/2 ΓR(s) L(s)=((0.983−0.178i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 189189    =    33⋅73^{3} \cdot 7
Sign: 0.983−0.178i0.983 - 0.178i
Analytic conductor: 0.8777120.877712
Root analytic conductor: 0.8777120.877712
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ189(4,⋅)\chi_{189} (4, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 189, (0: ), 0.983−0.178i)(1,\ 189,\ (0:\ ),\ 0.983 - 0.178i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.9005942956−0.08115069351i0.9005942956 - 0.08115069351i
L(12)L(\frac12) ≈\approx 0.9005942956−0.08115069351i0.9005942956 - 0.08115069351i
L(1)L(1) ≈\approx 0.8470139080+0.005783245469i0.8470139080 + 0.005783245469i
L(1)L(1) ≈\approx 0.8470139080+0.005783245469i0.8470139080 + 0.005783245469i

Euler product

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

Imaginary part of the first few zeros on the critical line

−26.958534491942302234800752659163, −26.45599898874385493568160035770, −25.27060154811276592911425897982, −24.847140511059930702792653210793, −23.32671965133622781117304664390, −22.008888313225996825647071813360, −21.40606365591722299779229729418, −20.38415099492951171654057223823, −19.23716421178072650654678393054, −18.48820174354523388409017190249, −17.65345954791622938995000943079, −16.67970905081524214577761442064, −15.75922968929549757010812897077, −14.32638154879259995039521660511, −13.45316412107960874251505567010, −11.930691515954167873007466156505, −11.13442926480575975858611737946, −10.084644610254845751940987731141, −9.20316269018797242099171900473, −8.162043322159255469533453887413, −6.721940748175658163115065172161, −6.07855528177532526035080015545, −3.93231796730081783775012416990, −2.64733202438969851383983833425, −1.42399009256563541854181268214, 1.16703385731665824040211567662, 2.36497961863503588172927467442, 4.4324072582864212231870218899, 5.88458385745003305389278040809, 6.631785483860637087597700653131, 8.15346134589651125769814334104, 8.94469513552205620575732347193, 9.92173532803487896809773181471, 10.86847931592323185300440072757, 12.202540578912384469413948597316, 13.34331136192323573140520882022, 14.5927920117360820980080588163, 15.59615365849984061545417112729, 16.62232596406221327518820185649, 17.55794003529621215486055612231, 18.019174271931716387803930438293, 19.49737631134906734449881814526, 20.17680263451330459171178803285, 21.1209403867970572384889256268, 22.2947496166154523612157822803, 23.67557998825979147366811960851, 24.45956471407013200408001685222, 25.462860251881397277301944027819, 25.87454024614980630564537379471, 27.17436715134841579494551203074

Graph of the ZZ-function along the critical line