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Properties

Label 2-189-63.61-c2-0-1
Degree $2$
Conductor $189$
Sign $0.731 + 0.681i$
Analytic cond. $5.14987$
Root an. cond. $2.26933$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.62 − 2.81i)2-s + (−3.29 + 5.70i)4-s + 3.39i·5-s + (−6.91 + 1.09i)7-s + 8.43·8-s + (9.56 − 5.52i)10-s + 18.7·11-s + (6.13 − 3.54i)13-s + (14.3 + 17.6i)14-s + (−0.537 − 0.931i)16-s + (5.22 − 3.01i)17-s + (22.2 + 12.8i)19-s + (−19.3 − 11.1i)20-s + (−30.5 − 52.9i)22-s − 1.98·23-s + ⋯
L(s)  = 1  + (−0.813 − 1.40i)2-s + (−0.823 + 1.42i)4-s + 0.678i·5-s + (−0.987 + 0.156i)7-s + 1.05·8-s + (0.956 − 0.552i)10-s + 1.70·11-s + (0.472 − 0.272i)13-s + (1.02 + 1.26i)14-s + (−0.0336 − 0.0582i)16-s + (0.307 − 0.177i)17-s + (1.16 + 0.674i)19-s + (−0.968 − 0.559i)20-s + (−1.38 − 2.40i)22-s − 0.0862·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 + 0.681i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.731 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.731 + 0.681i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ 0.731 + 0.681i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.840166 - 0.330783i\)
\(L(\frac12)\) \(\approx\) \(0.840166 - 0.330783i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (6.91 - 1.09i)T \)
good2 \( 1 + (1.62 + 2.81i)T + (-2 + 3.46i)T^{2} \)
5 \( 1 - 3.39iT - 25T^{2} \)
11 \( 1 - 18.7T + 121T^{2} \)
13 \( 1 + (-6.13 + 3.54i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-5.22 + 3.01i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-22.2 - 12.8i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + 1.98T + 529T^{2} \)
29 \( 1 + (12.9 - 22.4i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (6.84 + 3.95i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (19.4 - 33.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-42.7 + 24.6i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (18.6 - 32.3i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (27.1 - 15.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-36.6 - 63.3i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-45.9 - 26.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-34.8 + 20.1i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (38.1 - 66.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 17.9T + 5.04e3T^{2} \)
73 \( 1 + (-109. + 63.0i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (27.0 + 46.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (114. + 66.2i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (50.0 + 28.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (21.0 + 12.1i)T + (4.70e3 + 8.14e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.96205682642966043614649702020, −11.21905009421880392274706984994, −10.19162363223417143735437854876, −9.491302122048728646006938849751, −8.698161773994110116601042149275, −7.18366769962922489139057652959, −6.03660234689011447063602563409, −3.75243507153859551093395356977, −3.02062444117882931107776602903, −1.24781959038730935273114034272, 0.875249613534198076841377041621, 3.79212894707935157467673066510, 5.40674518302776671752134032867, 6.48177319423855697803728359192, 7.16182557678203464934883785317, 8.506313927980209708648925379408, 9.256852993403273768518942224910, 9.805760949244843940801518386785, 11.46159951274666036672088669973, 12.54823315255972629434005830307

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