↑

Properties

Label 2-189-7.6-c2-0-15
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

Related objects

Downloads

Learn more

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

\(L(\frac{3}{2})\) \(\approx\) \(3.94880 + 0.815183i\)
\(L(\frac12)\) \(\approx\) \(3.94880 + 0.815183i\)
\(L(2)\) 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 \( 1 + 133. iT - 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.74456329865988089405426039647, −11.38738047416401109968595384548, −10.85402846490691889195850590221, −10.04361692627411207089985279190, −7.49723359235621299745876571793, −7.01873311064069683272883613925, −6.00146167657473639187643369020, −4.73708845069415827466527751765, −3.32382865348583029768075793262, −2.72834826614591897096257297642, 1.98492604269335957197326592465, 3.56469576103795065299513564023, 4.94647938681446719808745960656, 5.38054698304026327269236624543, 6.55776733698316433229331395386, 8.103719055476080290612399611184, 9.183248990314405752490700943868, 10.71739397497412236818652081678, 11.92817648687657933707805884615, 12.58343347103967629948997825928

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