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Properties

Label 2-189-189.101-c1-0-13
Degree $2$
Conductor $189$
Sign $0.884 + 0.466i$
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  + (−1.63 + 1.94i)2-s + (1.01 − 1.40i)3-s + (−0.774 − 4.39i)4-s + (−0.356 + 0.299i)5-s + (1.07 + 4.26i)6-s + (−2.41 − 1.07i)7-s + (5.41 + 3.12i)8-s + (−0.937 − 2.84i)9-s − 1.18i·10-s + (1.58 − 1.89i)11-s + (−6.94 − 3.37i)12-s + (2.13 − 5.86i)13-s + (6.04 − 2.94i)14-s + (0.0577 + 0.804i)15-s + (−6.54 + 2.38i)16-s + 0.673·17-s + ⋯
L(s)  = 1  + (−1.15 + 1.37i)2-s + (0.586 − 0.810i)3-s + (−0.387 − 2.19i)4-s + (−0.159 + 0.133i)5-s + (0.438 + 1.74i)6-s + (−0.913 − 0.407i)7-s + (1.91 + 1.10i)8-s + (−0.312 − 0.949i)9-s − 0.374i·10-s + (0.479 − 0.571i)11-s + (−2.00 − 0.973i)12-s + (0.591 − 1.62i)13-s + (1.61 − 0.786i)14-s + (0.0149 + 0.207i)15-s + (−1.63 + 0.595i)16-s + 0.163·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.884 + 0.466i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.884 + 0.466i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.624097 - 0.154483i\)
\(L(\frac12)\) \(\approx\) \(0.624097 - 0.154483i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.01 + 1.40i)T \)
7 \( 1 + (2.41 + 1.07i)T \)
good2 \( 1 + (1.63 - 1.94i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (0.356 - 0.299i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-1.58 + 1.89i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-2.13 + 5.86i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 - 0.673T + 17T^{2} \)
19 \( 1 + 0.725iT - 19T^{2} \)
23 \( 1 + (1.10 - 3.04i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.84 + 5.07i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-6.82 + 1.20i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-3.70 + 6.41i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.54 + 1.29i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.41 - 8.03i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (1.58 - 8.98i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (11.5 + 6.66i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-10.2 - 3.72i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.919 - 0.162i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (0.0184 - 0.0154i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-8.62 + 4.98i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.49 - 1.43i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.843 - 0.707i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (4.97 - 1.80i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 10.2T + 89T^{2} \)
97 \( 1 + (-18.3 - 3.23i)T + (91.1 + 33.1i)T^{2} \)
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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

−12.89389731627496638632428495685, −11.27240020140588443777291350832, −9.981465561202087696815452550603, −9.241792312737284929210585126937, −8.119106939703902647163895574651, −7.60722517608766564499383613165, −6.46104998116779771475805105747, −5.82416188315470597971688964469, −3.35696595308704627538032206577, −0.812748914321913035796377257606, 2.06181851762054097184471879296, 3.38765281921919720642735055878, 4.39248590401810068391457898959, 6.69907165990263702626293508466, 8.311301983772390845846824788349, 8.950203582141904534876586355610, 9.727506458771030419843434888136, 10.35878798538511037113692672869, 11.57031623351475827138749416519, 12.18547346393328266590971525231

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