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Properties

Label 2-189-189.184-c1-0-5
Degree $2$
Conductor $189$
Sign $0.497 - 0.867i$
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  + (0.161 + 0.916i)2-s + (−1.65 + 0.499i)3-s + (1.06 − 0.387i)4-s + (0.260 − 1.47i)5-s + (−0.726 − 1.43i)6-s + (1.80 + 1.93i)7-s + (1.45 + 2.52i)8-s + (2.50 − 1.65i)9-s + 1.39·10-s + (0.663 + 3.76i)11-s + (−1.57 + 1.17i)12-s + (−0.747 − 0.627i)13-s + (−1.48 + 1.96i)14-s + (0.307 + 2.58i)15-s + (−0.341 + 0.286i)16-s − 3.53·17-s + ⋯
L(s)  = 1  + (0.114 + 0.648i)2-s + (−0.957 + 0.288i)3-s + (0.532 − 0.193i)4-s + (0.116 − 0.661i)5-s + (−0.296 − 0.587i)6-s + (0.680 + 0.732i)7-s + (0.515 + 0.892i)8-s + (0.833 − 0.552i)9-s + 0.442·10-s + (0.199 + 1.13i)11-s + (−0.454 + 0.339i)12-s + (−0.207 − 0.173i)13-s + (−0.397 + 0.524i)14-s + (0.0792 + 0.667i)15-s + (−0.0854 + 0.0717i)16-s − 0.858·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.497 - 0.867i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.497 - 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03769 + 0.601159i\)
\(L(\frac12)\) \(\approx\) \(1.03769 + 0.601159i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.65 - 0.499i)T \)
7 \( 1 + (-1.80 - 1.93i)T \)
good2 \( 1 + (-0.161 - 0.916i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (-0.260 + 1.47i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (-0.663 - 3.76i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (0.747 + 0.627i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + 3.53T + 17T^{2} \)
19 \( 1 - 4.19T + 19T^{2} \)
23 \( 1 + (4.86 + 4.08i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-1.19 + 1.00i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-5.51 + 2.00i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (3.65 + 6.32i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.42 + 6.22i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (10.7 + 3.92i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-2.28 - 0.832i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-0.809 - 1.40i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.26 + 1.06i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (14.1 + 5.14i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.199 - 1.12i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-2.88 + 5.00i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.29 - 5.71i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.390 - 2.21i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (0.529 - 0.443i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 + (-2.80 - 1.02i)T + (74.3 + 62.3i)T^{2} \)
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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

−12.29416435466558169236951120904, −11.96515790402116178578856692014, −10.85038419719628444188367312626, −9.861059229936196339912826771988, −8.624388192650093897891996939233, −7.36229397414178915890814103388, −6.35286733821117350037749930060, −5.25912745357847611736479991690, −4.63454358289214636204063385545, −1.90667285376926014350913090849, 1.46633684981639903980981413887, 3.25646784165248016719940101733, 4.72774848901488297895840987982, 6.29359006667058067300249149664, 7.02792420704333038608175040274, 8.105362931793960516889260700342, 10.05128674412450027085340078963, 10.67876244071702909294188135562, 11.58438118490246576680281889104, 11.84808431856020605108090425787

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