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Properties

Label 2-189-63.4-c3-0-5
Degree $2$
Conductor $189$
Sign $0.168 - 0.985i$
Analytic cond. $11.1513$
Root an. cond. $3.33936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.53 + 4.39i)2-s + (−8.90 − 15.4i)4-s − 18.4·5-s + (−9.68 − 15.7i)7-s + 49.8·8-s + (46.9 − 81.2i)10-s − 17.1·11-s + (−27.2 + 47.2i)13-s + (94.0 − 2.52i)14-s + (−55.3 + 95.7i)16-s + (7.61 − 13.1i)17-s + (43.7 + 75.7i)19-s + (164. + 284. i)20-s + (43.6 − 75.5i)22-s + 96.6·23-s + ⋯
L(s)  = 1  + (−0.898 + 1.55i)2-s + (−1.11 − 1.92i)4-s − 1.65·5-s + (−0.523 − 0.852i)7-s + 2.20·8-s + (1.48 − 2.56i)10-s − 0.470·11-s + (−0.582 + 1.00i)13-s + (1.79 − 0.0482i)14-s + (−0.864 + 1.49i)16-s + (0.108 − 0.188i)17-s + (0.527 + 0.914i)19-s + (1.83 + 3.18i)20-s + (0.422 − 0.732i)22-s + 0.876·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.168 - 0.985i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.168 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.168 - 0.985i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.168 - 0.985i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.329716 + 0.278241i\)
\(L(\frac12)\) \(\approx\) \(0.329716 + 0.278241i\)
\(L(\frac{5}{2})\) 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 \( 1 + (9.68 + 15.7i)T \)
good2 \( 1 + (2.53 - 4.39i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + 18.4T + 125T^{2} \)
11 \( 1 + 17.1T + 1.33e3T^{2} \)
13 \( 1 + (27.2 - 47.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-7.61 + 13.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-43.7 - 75.7i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 - 96.6T + 1.21e4T^{2} \)
29 \( 1 + (57.3 + 99.3i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (25.0 + 43.3i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-3.58 - 6.20i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-87.4 + 151. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (77.4 + 134. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (58.8 - 101. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-1.27 + 2.20i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-206. - 356. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-355. + 616. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (18.8 + 32.6i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 290.T + 3.57e5T^{2} \)
73 \( 1 + (-209. + 363. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-537. + 931. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-634. - 1.09e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (5.74 + 9.95i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-935. - 1.61e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.29027112828776113322571424410, −11.13315971146057132314753491128, −10.01476595559787330986403132514, −9.053586025217470454437095655770, −7.85440495523983023446941154247, −7.43950196590334871760636106961, −6.56497637909514955135377652834, −4.99203788096450329005700974337, −3.80213202452072819314715155099, −0.54695005347606480632954379475, 0.58016333136292782787072743946, 2.73864353153751440133342934668, 3.47646210670767390429027550618, 4.98922222456719636469220006764, 7.27398530474374708090914619130, 8.185095355054423550251827896734, 8.981055784238519013067127835722, 10.04135544525397769948287925550, 11.08340760010246221204746800880, 11.66917206262586381244401125868

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