Properties

Label 2-189-189.4-c1-0-17
Degree $2$
Conductor $189$
Sign $0.534 + 0.845i$
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  + (0.0570 + 0.323i)2-s + (−0.691 − 1.58i)3-s + (1.77 − 0.647i)4-s + (1.85 − 0.675i)5-s + (0.474 − 0.314i)6-s + (−2.33 − 1.24i)7-s + (0.639 + 1.10i)8-s + (−2.04 + 2.19i)9-s + (0.324 + 0.562i)10-s + (−3.32 − 1.21i)11-s + (−2.25 − 2.37i)12-s + (3.07 − 1.11i)13-s + (0.268 − 0.827i)14-s + (−2.35 − 2.48i)15-s + (2.57 − 2.16i)16-s + (3.33 + 5.78i)17-s + ⋯
L(s)  = 1  + (0.0403 + 0.228i)2-s + (−0.399 − 0.916i)3-s + (0.888 − 0.323i)4-s + (0.830 − 0.302i)5-s + (0.193 − 0.128i)6-s + (−0.883 − 0.469i)7-s + (0.226 + 0.391i)8-s + (−0.681 + 0.732i)9-s + (0.102 + 0.177i)10-s + (−1.00 − 0.365i)11-s + (−0.651 − 0.685i)12-s + (0.851 − 0.310i)13-s + (0.0717 − 0.221i)14-s + (−0.608 − 0.640i)15-s + (0.644 − 0.540i)16-s + (0.809 + 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.534 + 0.845i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.534 + 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14475 - 0.630612i\)
\(L(\frac12)\) \(\approx\) \(1.14475 - 0.630612i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.691 + 1.58i)T \)
7 \( 1 + (2.33 + 1.24i)T \)
good2 \( 1 + (-0.0570 - 0.323i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (-1.85 + 0.675i)T + (3.83 - 3.21i)T^{2} \)
11 \( 1 + (3.32 + 1.21i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (-3.07 + 1.11i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-3.33 - 5.78i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.31 + 5.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.04 - 5.94i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (4.03 + 1.46i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (3.97 - 1.44i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 - 2.07T + 37T^{2} \)
41 \( 1 + (0.663 - 0.241i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.66 - 9.42i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-7.26 - 2.64i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-0.108 + 0.188i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.08 + 2.58i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (5.32 + 1.93i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.695 + 3.94i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (4.16 - 7.21i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.42T + 73T^{2} \)
79 \( 1 + (-2.08 - 11.8i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (9.05 + 3.29i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-1.14 + 1.98i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.870 - 4.93i)T + (-91.1 + 33.1i)T^{2} \)
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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

−12.69235646137564039321790212215, −11.30813792063392227815651735510, −10.64707576731151664366351432935, −9.560311084401363363123936464763, −7.991528505656071766027127431786, −7.15343146745324202405147101829, −5.88684253386788862457047504174, −5.65729532377190058476629721486, −3.03130018723907266805105875044, −1.42511467139233583780883314926, 2.51663531793039641178806209813, 3.61502241887180977257719514633, 5.50748439447389212236627287692, 6.18294665118616607217306695327, 7.47142555998240068718507147148, 9.067955676623590066808208441116, 10.09015209269421200855542236159, 10.51450707754092673150516676728, 11.76221668835129175858638421479, 12.43593373881489286412285412406

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