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Properties

Label 1-189-189.184-r0-0-0
Degree $1$
Conductor $189$
Sign $0.545 + 0.838i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.545 + 0.838i$
Analytic conductor: \(0.877712\)
Root analytic conductor: \(0.877712\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (0:\ ),\ 0.545 + 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5144427261 + 0.2791130855i\)
\(L(\frac12)\) \(\approx\) \(0.5144427261 + 0.2791130855i\)
\(L(1)\) \(\approx\) \(0.6006522611 + 0.1314355681i\)
\(L(1)\) \(\approx\) \(0.6006522611 + 0.1314355681i\)

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

−26.9885264597529657019960106472, −26.32326454925432256351871370589, −25.30690771258804974359022491641, −24.252394311043475123911617080668, −23.17541043733494271196549476884, −22.15902554489442539153800162447, −20.86047623792057412722306789191, −20.219230586892338859677766084682, −19.09419278499188237650785708749, −18.482171446082792808421340029879, −17.51934018377438934936481500146, −16.10489705465507071220661799017, −15.75009679165892306597357496142, −14.40410977217697404844705775152, −12.77493132791118581961315758262, −11.98645263258063077126640292865, −10.79143174387981890378166793283, −10.23701105511327083215519058695, −8.67800195295663848465760867340, −7.87034711158664194138820342669, −7.027774811150753794459927024656, −5.41385503361266670946259417620, −3.56795433399936253462246625545, −2.73935755543480026201934584324, −0.74056409834025773810068304142, 1.22940375067293220935116064291, 2.94941246658944297121807866674, 4.61942918058541943115682726740, 5.90045455740010423381968663317, 7.405860239568665956514192510895, 7.895046344501581337369057722775, 9.15182008982845331635502133744, 10.10723206021266267385152531567, 11.40321663523085890895677475317, 12.06359753865536318250348636787, 13.63683142537748918555345857386, 14.97135655049408693991920043235, 15.81232434128355591041715657759, 16.50151279725860415837727062003, 17.60160756031900609992598107667, 18.758698122643996233614294941933, 19.28867932493650579462633087577, 20.46779214404963950743530696384, 21.12959458965818261799097697836, 22.919201958308423557005558418124, 23.686147280655010037986167862427, 24.40995966823279463209489645732, 25.58261864158159988431816720816, 26.4296265519772798177573868871, 27.18628859654029698990759131280

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