Properties

Label 2-189-63.11-c2-0-8
Degree $2$
Conductor $189$
Sign $0.487 + 0.873i$
Analytic cond. $5.14987$
Root an. cond. $2.26933$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.03i·2-s + 2.93·4-s + (2.10 + 1.21i)5-s + (−4.75 − 5.13i)7-s − 7.16i·8-s + (1.25 − 2.17i)10-s + (11.6 − 6.69i)11-s + (5.06 + 8.78i)13-s + (−5.31 + 4.92i)14-s + 4.30·16-s + (18.6 + 10.7i)17-s + (−10.4 − 18.0i)19-s + (6.16 + 3.55i)20-s + (−6.92 − 12.0i)22-s + (−14.5 − 8.41i)23-s + ⋯
L(s)  = 1  − 0.517i·2-s + 0.732·4-s + (0.420 + 0.242i)5-s + (−0.679 − 0.733i)7-s − 0.896i·8-s + (0.125 − 0.217i)10-s + (1.05 − 0.609i)11-s + (0.389 + 0.675i)13-s + (−0.379 + 0.351i)14-s + 0.269·16-s + (1.09 + 0.633i)17-s + (−0.548 − 0.950i)19-s + (0.308 + 0.177i)20-s + (−0.314 − 0.545i)22-s + (−0.634 − 0.366i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.487 + 0.873i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ 0.487 + 0.873i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.63764 - 0.961288i\)
\(L(\frac12)\) \(\approx\) \(1.63764 - 0.961288i\)
\(L(2)\) 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 \( 1 + (4.75 + 5.13i)T \)
good2 \( 1 + 1.03iT - 4T^{2} \)
5 \( 1 + (-2.10 - 1.21i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-11.6 + 6.69i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-5.06 - 8.78i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-18.6 - 10.7i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (10.4 + 18.0i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (14.5 + 8.41i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-8.10 - 4.67i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 12.1T + 961T^{2} \)
37 \( 1 + (-22.0 - 38.2i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (43.0 - 24.8i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (25.8 - 44.7i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 - 63.4iT - 2.20e3T^{2} \)
53 \( 1 + (46.1 + 26.6i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + 12.7iT - 3.48e3T^{2} \)
61 \( 1 - 60.6T + 3.72e3T^{2} \)
67 \( 1 + 78.7T + 4.48e3T^{2} \)
71 \( 1 + 9.91iT - 5.04e3T^{2} \)
73 \( 1 + (49.3 - 85.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 - 53.3T + 6.24e3T^{2} \)
83 \( 1 + (-81.5 - 47.0i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (17.1 - 9.89i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (35.2 - 60.9i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−12.00847415421230948587493438677, −11.20186949852431170866120076125, −10.25318256264433683673480057756, −9.544631702911046962382424983145, −8.098825061940145981144179105780, −6.51590965280360172594112208062, −6.37747762387657543592365857895, −4.11685148677834691629629643111, −3.00000166815192582640446184418, −1.30917989937077516268672833149, 1.87743503561457455007620360691, 3.47006481195178798507069393101, 5.46664528565064457972795682939, 6.13605262836767799464591417795, 7.23410715637299078453024476055, 8.376567343966551196408634016756, 9.516732578986788555274948947080, 10.37442354088351538234869278789, 11.87261173419397797101621913184, 12.23310109853834910448372231505

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