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Properties

Label 2-189-9.4-c1-0-3
Degree $2$
Conductor $189$
Sign $0.939 - 0.342i$
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  + (0.673 + 1.16i)2-s + (0.0923 − 0.160i)4-s + (1.26 − 2.19i)5-s + (−0.5 − 0.866i)7-s + 2.94·8-s + 3.41·10-s + (0.233 + 0.405i)11-s + (−2.91 + 5.04i)13-s + (0.673 − 1.16i)14-s + (1.79 + 3.11i)16-s − 3.87·17-s − 2.18·19-s + (−0.233 − 0.405i)20-s + (−0.315 + 0.545i)22-s + (−0.0530 + 0.0918i)23-s + ⋯
L(s)  = 1  + (0.476 + 0.825i)2-s + (0.0461 − 0.0800i)4-s + (0.566 − 0.980i)5-s + (−0.188 − 0.327i)7-s + 1.04·8-s + 1.07·10-s + (0.0705 + 0.122i)11-s + (−0.807 + 1.39i)13-s + (0.180 − 0.311i)14-s + (0.449 + 0.778i)16-s − 0.940·17-s − 0.501·19-s + (−0.0523 − 0.0906i)20-s + (−0.0672 + 0.116i)22-s + (−0.0110 + 0.0191i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62728 + 0.286934i\)
\(L(\frac12)\) \(\approx\) \(1.62728 + 0.286934i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.673 - 1.16i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.26 + 2.19i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.233 - 0.405i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.91 - 5.04i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.87T + 17T^{2} \)
19 \( 1 + 2.18T + 19T^{2} \)
23 \( 1 + (0.0530 - 0.0918i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.39 - 7.60i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.84 + 6.65i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.68T + 37T^{2} \)
41 \( 1 + (1.11 - 1.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.613 + 1.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.66 + 4.61i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.716T + 53T^{2} \)
59 \( 1 + (-0.368 + 0.637i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.479 + 0.829i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.81 + 8.34i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 + (-6.31 - 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.36 + 2.36i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 8.11T + 89T^{2} \)
97 \( 1 + (-6.80 - 11.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.94067948780745444383563814491, −11.82633763926628908440874204430, −10.56203415398815050158729204321, −9.523639066865171147704973217939, −8.580480771207190432472562171601, −7.11933639632968488614605348758, −6.40420101292222859163302143262, −5.08339352163665890410551723168, −4.37330280820422888416478285161, −1.83261589329923896951674637946, 2.35125114336309548659765692959, 3.15142591218481922883376292724, 4.73892170285543167822416058950, 6.18851063398547620558297262052, 7.26152307067416922619933753364, 8.502024561904440908399104907881, 10.14543836386220117513996008643, 10.47495256852791659497306067099, 11.60794913630626831799233714036, 12.48234085862372157859011272148

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