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Properties

Label 1-189-189.23-r1-0-0
Degree $1$
Conductor $189$
Sign $-0.891 - 0.453i$
Analytic cond. $20.3108$
Root an. cond. $20.3108$
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  + (−0.173 + 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.173 − 0.984i)5-s + (0.5 − 0.866i)8-s + 10-s + (−0.173 + 0.984i)11-s + (0.766 − 0.642i)13-s + (0.766 + 0.642i)16-s − 17-s + 19-s + (−0.173 + 0.984i)20-s + (−0.939 − 0.342i)22-s + (−0.766 + 0.642i)23-s + (−0.939 + 0.342i)25-s + (0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.173 − 0.984i)5-s + (0.5 − 0.866i)8-s + 10-s + (−0.173 + 0.984i)11-s + (0.766 − 0.642i)13-s + (0.766 + 0.642i)16-s − 17-s + 19-s + (−0.173 + 0.984i)20-s + (−0.939 − 0.342i)22-s + (−0.766 + 0.642i)23-s + (−0.939 + 0.342i)25-s + (0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.891 - 0.453i$
Analytic conductor: \(20.3108\)
Root analytic conductor: \(20.3108\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (1:\ ),\ -0.891 - 0.453i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02912768128 + 0.1214914835i\)
\(L(\frac12)\) \(\approx\) \(0.02912768128 + 0.1214914835i\)
\(L(1)\) \(\approx\) \(0.6543239911 + 0.2184118748i\)
\(L(1)\) \(\approx\) \(0.6543239911 + 0.2184118748i\)

Euler product

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

−26.52158732908493626291948562356, −25.88147380685461623036310879034, −24.2439610035334145584617072047, −23.26841862748351539189400731540, −22.18339509355179448383028445867, −21.737511213749866966539669229359, −20.500814727969260497008239613497, −19.61390164682403759522656818192, −18.433019837916298721488164902571, −18.288872253640332229149416340676, −16.756016237888746513319610723693, −15.61855443883055052500682783271, −14.12692956142728338710544881502, −13.66267307553531816179139755584, −12.247773713954143641452772558155, −11.12362435650521080599179456046, −10.73456523840227284272079918814, −9.344046886068861373326581020879, −8.36631546032035180358111126491, −7.038983224107822799174781795680, −5.64558033970740818547801627272, −4.007809740570178515585915962500, −3.13913755309338836671563706694, −1.83706675829212937751774723235, −0.04732630512368813307922957749, 1.50843536061875089092958909354, 3.83624817654899732296578564675, 4.925247634274487231257883719050, 5.86151432703939330274609449567, 7.265728687755029994276953833327, 8.17319118294334721793376218206, 9.17189745171827130734052834248, 10.1002860177375852125340840677, 11.72209045610747491417266392669, 13.02223345368267631120143738420, 13.5870083902973235547020167872, 15.13851639679371495818733538687, 15.70935993584872195721177657236, 16.69713187638314430054249165040, 17.67541661096323602671399926070, 18.38888136353733749758941086160, 19.88379036916631792345600195595, 20.49842501487196861563501114841, 21.98676807022375105693525123810, 22.940263875876864668951279291267, 23.81211193108767895940486417152, 24.62210861223621464751299970480, 25.44377397678973426054331678755, 26.31660554169309115484650319186, 27.47927083430945464219768433392

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