Properties

Label 2-189-27.4-c1-0-5
Degree $2$
Conductor $189$
Sign $0.858 + 0.513i$
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  + (−2.52 − 0.919i)2-s + (1.69 + 0.342i)3-s + (4.00 + 3.35i)4-s + (0.203 − 1.15i)5-s + (−3.97 − 2.42i)6-s + (0.766 − 0.642i)7-s + (−4.33 − 7.50i)8-s + (2.76 + 1.16i)9-s + (−1.57 + 2.72i)10-s + (0.971 + 5.51i)11-s + (5.64 + 7.07i)12-s + (−0.292 + 0.106i)13-s + (−2.52 + 0.919i)14-s + (0.739 − 1.88i)15-s + (2.22 + 12.6i)16-s + (3.00 − 5.20i)17-s + ⋯
L(s)  = 1  + (−1.78 − 0.650i)2-s + (0.980 + 0.197i)3-s + (2.00 + 1.67i)4-s + (0.0908 − 0.515i)5-s + (−1.62 − 0.990i)6-s + (0.289 − 0.242i)7-s + (−1.53 − 2.65i)8-s + (0.921 + 0.387i)9-s + (−0.497 + 0.861i)10-s + (0.292 + 1.66i)11-s + (1.62 + 2.04i)12-s + (−0.0812 + 0.0295i)13-s + (−0.674 + 0.245i)14-s + (0.190 − 0.487i)15-s + (0.557 + 3.16i)16-s + (0.728 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 + 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.858 + 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.858 + 0.513i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.858 + 0.513i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.767514 - 0.211960i\)
\(L(\frac12)\) \(\approx\) \(0.767514 - 0.211960i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.69 - 0.342i)T \)
7 \( 1 + (-0.766 + 0.642i)T \)
good2 \( 1 + (2.52 + 0.919i)T + (1.53 + 1.28i)T^{2} \)
5 \( 1 + (-0.203 + 1.15i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (-0.971 - 5.51i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (0.292 - 0.106i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-3.00 + 5.20i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.92 + 3.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.48 - 1.24i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (0.141 + 0.0514i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (4.98 + 4.17i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (-1.20 + 2.08i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.879 + 0.320i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.356 - 2.02i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (0.0877 - 0.0736i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + 12.9T + 53T^{2} \)
59 \( 1 + (2.31 - 13.1i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (4.94 - 4.14i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (6.91 - 2.51i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-1.70 + 2.95i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.48 - 2.57i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.0 + 3.64i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (14.6 + 5.33i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (3.11 + 5.38i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.36 + 7.73i)T + (-91.1 + 33.1i)T^{2} \)
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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

−12.29266246072268779939883836177, −11.19863390753688962251282949438, −10.09114782247574792657863122292, −9.405238044120409729481561675219, −8.852948417794311280984489856067, −7.54651585481985535295922215953, −7.18893185385281680731179055454, −4.51097189870244744267875465781, −2.81304045352120544714283485408, −1.53992835954630780785411287278, 1.54347909735362424553852635191, 3.17401401964878539031019655220, 5.90878717278655140919415474438, 6.76215892120636487899312530107, 8.043258175456658181392427365423, 8.420763902917371918632455872560, 9.353536583977864185660381972938, 10.44547833233489482908745751533, 11.07029907799855384998702190051, 12.54571255143420151831903698552

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