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Properties

Label 2-189-63.41-c1-0-5
Degree $2$
Conductor $189$
Sign $0.412 + 0.911i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.97 − 1.13i)2-s + (1.59 − 2.75i)4-s + (0.717 − 1.24i)5-s + (−2.40 + 1.11i)7-s − 2.69i·8-s − 3.26i·10-s + (−2.80 + 1.61i)11-s + (4.43 + 2.55i)13-s + (−3.47 + 4.92i)14-s + (0.119 + 0.207i)16-s − 1.09·17-s − 4.48i·19-s + (−2.28 − 3.95i)20-s + (−3.68 + 6.37i)22-s + (−3.47 − 2.00i)23-s + ⋯
L(s)  = 1  + (1.39 − 0.804i)2-s + (0.795 − 1.37i)4-s + (0.320 − 0.555i)5-s + (−0.907 + 0.419i)7-s − 0.951i·8-s − 1.03i·10-s + (−0.844 + 0.487i)11-s + (1.22 + 0.709i)13-s + (−0.927 + 1.31i)14-s + (0.0298 + 0.0517i)16-s − 0.264·17-s − 1.02i·19-s + (−0.510 − 0.883i)20-s + (−0.784 + 1.35i)22-s + (−0.723 − 0.417i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.412 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.412 + 0.911i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.412 + 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87647 - 1.21081i\)
\(L(\frac12)\) \(\approx\) \(1.87647 - 1.21081i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.40 - 1.11i)T \)
good2 \( 1 + (-1.97 + 1.13i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-0.717 + 1.24i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.80 - 1.61i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.43 - 2.55i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.09T + 17T^{2} \)
19 \( 1 + 4.48iT - 19T^{2} \)
23 \( 1 + (3.47 + 2.00i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.02 - 0.593i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.24 + 1.87i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.239T + 37T^{2} \)
41 \( 1 + (3.71 - 6.43i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.82 + 6.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.11 - 3.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 7.01iT - 53T^{2} \)
59 \( 1 + (-4.73 + 8.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.82 - 1.63i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.330 - 0.571i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.82iT - 71T^{2} \)
73 \( 1 + 7.31iT - 73T^{2} \)
79 \( 1 + (1.83 + 3.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.45 - 9.44i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 13.6T + 89T^{2} \)
97 \( 1 + (-2.69 + 1.55i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.68162318850598966435733523663, −11.64713067064845463435717163028, −10.76095303310469766563008151195, −9.628600098031014269397941764253, −8.575447166522664524252230288301, −6.71179010808577064339063086212, −5.69333478496586048373279816232, −4.68644442796253833564957911446, −3.42815671689783544475660044488, −2.08174355276751018544557511780, 3.04335907719612786680512741462, 3.92614880930788595164598737937, 5.60649272050962048466644119916, 6.17006687001346298839721624941, 7.22171907409034443890066274327, 8.333849557911435724737271464726, 10.02866191522700886308357888256, 10.81634194285049476698269939668, 12.22470399960516451313369601194, 13.15952723755064175762784244525

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