Properties

Label 2-189-27.7-c1-0-8
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: $\chi_{189} (169, \cdot )$
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

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

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