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Properties

Label 2-189-189.131-c1-0-3
Degree $2$
Conductor $189$
Sign $-0.213 + 0.976i$
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  + (−1.60 − 1.91i)2-s + (−1.44 + 0.952i)3-s + (−0.733 + 4.16i)4-s + (−0.273 − 0.229i)5-s + (4.14 + 1.23i)6-s + (2.61 + 0.393i)7-s + (4.81 − 2.77i)8-s + (1.18 − 2.75i)9-s + 0.892i·10-s + (−2.21 − 2.63i)11-s + (−2.90 − 6.71i)12-s + (−0.362 − 0.995i)13-s + (−3.44 − 5.63i)14-s + (0.615 + 0.0714i)15-s + (−5.08 − 1.85i)16-s + 5.82·17-s + ⋯
L(s)  = 1  + (−1.13 − 1.35i)2-s + (−0.835 + 0.550i)3-s + (−0.366 + 2.08i)4-s + (−0.122 − 0.102i)5-s + (1.69 + 0.504i)6-s + (0.988 + 0.148i)7-s + (1.70 − 0.982i)8-s + (0.394 − 0.918i)9-s + 0.282i·10-s + (−0.667 − 0.795i)11-s + (−0.838 − 1.93i)12-s + (−0.100 − 0.276i)13-s + (−0.920 − 1.50i)14-s + (0.158 + 0.0184i)15-s + (−1.27 − 0.462i)16-s + 1.41·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.213 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.213 + 0.976i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.213 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.318734 - 0.396084i\)
\(L(\frac12)\) \(\approx\) \(0.318734 - 0.396084i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.44 - 0.952i)T \)
7 \( 1 + (-2.61 - 0.393i)T \)
good2 \( 1 + (1.60 + 1.91i)T + (-0.347 + 1.96i)T^{2} \)
5 \( 1 + (0.273 + 0.229i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (2.21 + 2.63i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (0.362 + 0.995i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 - 5.82T + 17T^{2} \)
19 \( 1 + 4.08iT - 19T^{2} \)
23 \( 1 + (-0.737 - 2.02i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-2.91 + 7.99i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (-3.71 - 0.655i)T + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (-0.937 - 1.62i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.24 + 1.54i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.24 + 7.08i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-2.12 - 12.0i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (7.91 - 4.56i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.87 - 1.41i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-13.0 + 2.30i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (-3.53 - 2.96i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (7.24 + 4.18i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.45 + 3.72i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.2 - 8.62i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-3.80 - 1.38i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 + (-4.45 + 0.785i)T + (91.1 - 33.1i)T^{2} \)
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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

−11.82742856595989492301584505654, −11.20989146672668147925600962680, −10.43633565350735687243658102009, −9.660194079163534824013749499160, −8.492204822837719763515926361997, −7.69388069299922639133595511613, −5.71803383026809459111761843965, −4.38941288305934384581787914350, −2.86482334881823590306374278603, −0.852340565517255519102660794241, 1.41978014533203601293627758965, 4.90256895098033179794704415660, 5.69580929056151017250369043353, 6.97815347251006639836427368567, 7.64547765769815080801895968325, 8.360280200150922426349765006666, 9.886660634107762554339706745118, 10.55733226859129629475440781804, 11.71519659021429470056293493448, 12.79590332530387955443219980764

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