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Properties

Label 2-189-7.6-c2-0-3
Degree $2$
Conductor $189$
Sign $-0.918 - 0.396i$
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  − 3.71·2-s + 9.77·4-s + 8.26i·5-s + (−2.77 + 6.42i)7-s − 21.4·8-s − 30.6i·10-s + 10.2·11-s + 6.42i·13-s + (10.2 − 23.8i)14-s + 40.4·16-s − 15.5i·17-s + 4.96i·19-s + 80.7i·20-s − 38.1·22-s − 28.8·23-s + ⋯
L(s)  = 1  − 1.85·2-s + 2.44·4-s + 1.65i·5-s + (−0.396 + 0.918i)7-s − 2.67·8-s − 3.06i·10-s + 0.935·11-s + 0.494i·13-s + (0.734 − 1.70i)14-s + 2.52·16-s − 0.916i·17-s + 0.261i·19-s + 4.03i·20-s − 1.73·22-s − 1.25·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 - 0.396i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.918 - 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.918 - 0.396i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ -0.918 - 0.396i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0887716 + 0.430015i\)
\(L(\frac12)\) \(\approx\) \(0.0887716 + 0.430015i\)
\(L(2)\) 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 \( 1 + (2.77 - 6.42i)T \)
good2 \( 1 + 3.71T + 4T^{2} \)
5 \( 1 - 8.26iT - 25T^{2} \)
11 \( 1 - 10.2T + 121T^{2} \)
13 \( 1 - 6.42iT - 169T^{2} \)
17 \( 1 + 15.5iT - 289T^{2} \)
19 \( 1 - 4.96iT - 361T^{2} \)
23 \( 1 + 28.8T + 529T^{2} \)
29 \( 1 - 2.01T + 841T^{2} \)
31 \( 1 - 16.3iT - 961T^{2} \)
37 \( 1 + 33.2T + 1.36e3T^{2} \)
41 \( 1 - 39.4iT - 1.68e3T^{2} \)
43 \( 1 + 3.91T + 1.84e3T^{2} \)
47 \( 1 - 41.3iT - 2.20e3T^{2} \)
53 \( 1 - 43.6T + 2.80e3T^{2} \)
59 \( 1 - 30.2iT - 3.48e3T^{2} \)
61 \( 1 + 107. iT - 3.72e3T^{2} \)
67 \( 1 + 36.5T + 4.48e3T^{2} \)
71 \( 1 + 79.1T + 5.04e3T^{2} \)
73 \( 1 + 38.5iT - 5.32e3T^{2} \)
79 \( 1 - 100.T + 6.24e3T^{2} \)
83 \( 1 + 52.2iT - 6.88e3T^{2} \)
89 \( 1 + 71.5iT - 7.92e3T^{2} \)
97 \( 1 - 133. iT - 9.40e3T^{2} \)
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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

−12.00712150430277518512029303433, −11.55404539911199069752279515265, −10.52397376065892994238344844046, −9.718862402246554221236476779955, −8.969153749770512470678754196040, −7.74725796396564843665219945735, −6.75211889038851559956340613113, −6.18615679470758484101418915812, −3.20420595929600706155917288721, −2.05328149151441093746675206214, 0.46437153256673815461347379093, 1.61769852783196187758643972652, 3.99960533388458137407368981477, 5.88460093761236578110275127905, 7.12816009955212791441405660198, 8.210193554481988301984239927065, 8.842293912453198924667614958628, 9.763984724308124729878147899630, 10.49563774599423597333918474435, 11.75591469687020714807487560453

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