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Properties

Label 2-189-63.59-c1-0-5
Degree $2$
Conductor $189$
Sign $-0.464 + 0.885i$
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  + (0.254 + 0.146i)2-s + (−0.956 − 1.65i)4-s − 3.06·5-s + (−1.22 − 2.34i)7-s − 1.15i·8-s + (−0.778 − 0.449i)10-s − 3.89i·11-s + (2.02 + 1.17i)13-s + (0.0325 − 0.776i)14-s + (−1.74 + 3.02i)16-s + (−1.68 + 2.91i)17-s + (2.20 − 1.27i)19-s + (2.92 + 5.07i)20-s + (0.572 − 0.991i)22-s − 2.98i·23-s + ⋯
L(s)  = 1  + (0.179 + 0.103i)2-s + (−0.478 − 0.828i)4-s − 1.36·5-s + (−0.463 − 0.886i)7-s − 0.406i·8-s + (−0.246 − 0.142i)10-s − 1.17i·11-s + (0.562 + 0.324i)13-s + (0.00869 − 0.207i)14-s + (−0.436 + 0.755i)16-s + (−0.408 + 0.706i)17-s + (0.506 − 0.292i)19-s + (0.654 + 1.13i)20-s + (0.122 − 0.211i)22-s − 0.621i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.464 + 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.464 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.464 + 0.885i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.464 + 0.885i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.358548 - 0.593242i\)
\(L(\frac12)\) \(\approx\) \(0.358548 - 0.593242i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (1.22 + 2.34i)T \)
good2 \( 1 + (-0.254 - 0.146i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 3.06T + 5T^{2} \)
11 \( 1 + 3.89iT - 11T^{2} \)
13 \( 1 + (-2.02 - 1.17i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.68 - 2.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.20 + 1.27i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.98iT - 23T^{2} \)
29 \( 1 + (-3.67 + 2.12i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.409 - 0.236i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.89 + 6.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.12 + 5.41i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.06 - 3.57i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.02 + 3.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.99 - 2.88i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.34 - 4.05i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.38 + 0.800i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.787 + 1.36i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.6iT - 71T^{2} \)
73 \( 1 + (0.856 + 0.494i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.63 + 8.03i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.49 - 9.51i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.15 + 3.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.98 - 2.87i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.27027417117462452867336680791, −11.03252608691680407090038686469, −10.57712838320788561075205256651, −9.147141640706988410879058616278, −8.235815452478610872143369712098, −6.99390744133544178512423090293, −5.93070708599255846371912494166, −4.37370206069095027054862632734, −3.60483737371576070938052427901, −0.60589130754647243049971770521, 2.91865232577819247315551420581, 4.01699845085200827681155761476, 5.14380886110466720453810277081, 6.93839010574000485039081086444, 7.87923907703673205032975100521, 8.737650986747652898974902665659, 9.765031655112591842271759118751, 11.35951682406637918104780657365, 12.04206297948414671372266109906, 12.61497720320669124132583728367

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