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Properties

Label 2-189-63.25-c3-0-21
Degree $2$
Conductor $189$
Sign $-0.919 + 0.393i$
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  + 2.93·2-s + 0.611·4-s + (−1.84 + 3.20i)5-s + (−18.1 − 3.79i)7-s − 21.6·8-s + (−5.42 + 9.39i)10-s + (−32.8 − 56.9i)11-s + (2.73 + 4.74i)13-s + (−53.1 − 11.1i)14-s − 68.5·16-s + (25.1 − 43.5i)17-s + (0.769 + 1.33i)19-s + (−1.13 + 1.95i)20-s + (−96.4 − 166. i)22-s + (−60.0 + 103. i)23-s + ⋯
L(s)  = 1  + 1.03·2-s + 0.0764·4-s + (−0.165 + 0.286i)5-s + (−0.978 − 0.204i)7-s − 0.958·8-s + (−0.171 + 0.297i)10-s + (−0.900 − 1.55i)11-s + (0.0584 + 0.101i)13-s + (−1.01 − 0.212i)14-s − 1.07·16-s + (0.358 − 0.621i)17-s + (0.00929 + 0.0161i)19-s + (−0.0126 + 0.0218i)20-s + (−0.934 − 1.61i)22-s + (−0.544 + 0.942i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 + 0.393i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.919 + 0.393i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.919 + 0.393i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.103278 - 0.503697i\)
\(L(\frac12)\) \(\approx\) \(0.103278 - 0.503697i\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.1 + 3.79i)T \)
good2 \( 1 - 2.93T + 8T^{2} \)
5 \( 1 + (1.84 - 3.20i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (32.8 + 56.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-2.73 - 4.74i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-25.1 + 43.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-0.769 - 1.33i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (60.0 - 103. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-39.2 + 68.0i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 303.T + 2.97e4T^{2} \)
37 \( 1 + (-96.6 - 167. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (196. + 340. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (138. - 239. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 252.T + 1.03e5T^{2} \)
53 \( 1 + (-204. + 353. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + 262.T + 2.05e5T^{2} \)
61 \( 1 + 112.T + 2.26e5T^{2} \)
67 \( 1 + 98.2T + 3.00e5T^{2} \)
71 \( 1 - 255.T + 3.57e5T^{2} \)
73 \( 1 + (-344. + 596. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + 1.08e3T + 4.93e5T^{2} \)
83 \( 1 + (152. - 263. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (550. + 953. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-493. + 855. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.84804719339111521417091778435, −10.93345051077926251497382416567, −9.725407968515751134174982143630, −8.692498838291200521033885544346, −7.33131170269102511763855559794, −6.05345244277574062969943637052, −5.29820768879669283085180755219, −3.66947282019483636308205708223, −3.01313028942995703685058384380, −0.15265305050047434167325565589, 2.51737920597501027807488716585, 3.89001797448711359448625337897, 4.92795365730720023261313005517, 5.99297020769186827184946873005, 7.16417880620029673335663070421, 8.552809304962831304949133048407, 9.645920619665779856853881420474, 10.53649712375024941656625165219, 12.22638071304523111804369670442, 12.56708773086337046813857476138

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