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Properties

Label 2-189-63.16-c1-0-4
Degree $2$
Conductor $189$
Sign $0.649 + 0.760i$
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  + (−0.247 − 0.429i)2-s + (0.877 − 1.51i)4-s + 3.69·5-s + (−2.60 + 0.436i)7-s − 1.86·8-s + (−0.915 − 1.58i)10-s + 0.892·11-s + (0.598 + 1.03i)13-s + (0.834 + 1.01i)14-s + (−1.29 − 2.23i)16-s + (0.124 + 0.216i)17-s + (1.40 − 2.43i)19-s + (3.23 − 5.60i)20-s + (−0.221 − 0.383i)22-s − 2.47·23-s + ⋯
L(s)  = 1  + (−0.175 − 0.303i)2-s + (0.438 − 0.759i)4-s + 1.65·5-s + (−0.986 + 0.165i)7-s − 0.658·8-s + (−0.289 − 0.501i)10-s + 0.269·11-s + (0.165 + 0.287i)13-s + (0.223 + 0.270i)14-s + (−0.323 − 0.559i)16-s + (0.0303 + 0.0525i)17-s + (0.322 − 0.557i)19-s + (0.724 − 1.25i)20-s + (−0.0471 − 0.0817i)22-s − 0.516·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.649 + 0.760i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.649 + 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21992 - 0.562784i\)
\(L(\frac12)\) \(\approx\) \(1.21992 - 0.562784i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.60 - 0.436i)T \)
good2 \( 1 + (0.247 + 0.429i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 3.69T + 5T^{2} \)
11 \( 1 - 0.892T + 11T^{2} \)
13 \( 1 + (-0.598 - 1.03i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.124 - 0.216i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.40 + 2.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.47T + 23T^{2} \)
29 \( 1 + (2.07 - 3.58i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.79 - 3.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.36 - 4.09i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.39 - 4.14i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.98 - 8.64i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.08 + 8.81i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.94 - 8.56i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.906 + 1.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.40 + 9.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.514 - 0.891i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.94T + 71T^{2} \)
73 \( 1 + (0.915 + 1.58i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.899 - 1.55i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.16 - 10.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.20 + 2.08i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.52 + 9.56i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.48663365437221006033014924863, −11.27714767249515794304519121817, −10.20587198856878350696004547745, −9.657134190365855193221180249026, −8.924589258569616983561227533355, −6.80920788397647312229814803355, −6.18711215482354312833072563765, −5.22656091812064285484311751527, −2.96862605651209327186786161768, −1.65675695120095287667975052762, 2.24135452754710516531159792381, 3.61048758705014705174472235425, 5.70485340535027447940023620805, 6.37527412748882336323022456510, 7.44400527610162063914174229300, 8.832226976831569601213186523300, 9.661801132014108334775514310630, 10.49444162838935184572326035110, 11.88313669222508406344804892478, 12.88093595272550887487430027594

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