↑

Properties

Label 2-264-88.21-c2-0-3
Degree $2$
Conductor $264$
Sign $0.476 - 0.879i$
Analytic cond. $7.19347$
Root an. cond. $2.68206$
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  + (−0.387 − 1.96i)2-s − 1.73i·3-s + (−3.69 + 1.52i)4-s − 0.655i·5-s + (−3.39 + 0.671i)6-s + 3.90i·7-s + (4.41 + 6.66i)8-s − 2.99·9-s + (−1.28 + 0.253i)10-s + (−9.70 + 5.17i)11-s + (2.63 + 6.40i)12-s − 18.7·13-s + (7.66 − 1.51i)14-s − 1.13·15-s + (11.3 − 11.2i)16-s + 30.1i·17-s + ⋯
L(s)  = 1  + (−0.193 − 0.981i)2-s − 0.577i·3-s + (−0.924 + 0.380i)4-s − 0.131i·5-s + (−0.566 + 0.111i)6-s + 0.557i·7-s + (0.552 + 0.833i)8-s − 0.333·9-s + (−0.128 + 0.0253i)10-s + (−0.882 + 0.470i)11-s + (0.219 + 0.533i)12-s − 1.44·13-s + (0.547 − 0.108i)14-s − 0.0756·15-s + (0.710 − 0.703i)16-s + 1.77i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.476 - 0.879i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.476 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $0.476 - 0.879i$
Analytic conductor: \(7.19347\)
Root analytic conductor: \(2.68206\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{264} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 264,\ (\ :1),\ 0.476 - 0.879i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.358348 + 0.213491i\)
\(L(\frac12)\) \(\approx\) \(0.358348 + 0.213491i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.387 + 1.96i)T \)
3 \( 1 + 1.73iT \)
11 \( 1 + (9.70 - 5.17i)T \)
good5 \( 1 + 0.655iT - 25T^{2} \)
7 \( 1 - 3.90iT - 49T^{2} \)
13 \( 1 + 18.7T + 169T^{2} \)
17 \( 1 - 30.1iT - 289T^{2} \)
19 \( 1 + 8.44T + 361T^{2} \)
23 \( 1 - 13.1T + 529T^{2} \)
29 \( 1 - 10.4T + 841T^{2} \)
31 \( 1 + 30.9T + 961T^{2} \)
37 \( 1 + 28.6iT - 1.36e3T^{2} \)
41 \( 1 - 0.794iT - 1.68e3T^{2} \)
43 \( 1 + 27.8T + 1.84e3T^{2} \)
47 \( 1 + 72.9T + 2.20e3T^{2} \)
53 \( 1 - 52.5iT - 2.80e3T^{2} \)
59 \( 1 - 84.2iT - 3.48e3T^{2} \)
61 \( 1 + 21.7T + 3.72e3T^{2} \)
67 \( 1 + 103. iT - 4.48e3T^{2} \)
71 \( 1 + 117.T + 5.04e3T^{2} \)
73 \( 1 - 106. iT - 5.32e3T^{2} \)
79 \( 1 + 56.8iT - 6.24e3T^{2} \)
83 \( 1 + 61.3T + 6.88e3T^{2} \)
89 \( 1 + 58.5T + 7.92e3T^{2} \)
97 \( 1 + 27.6T + 9.40e3T^{2} \)
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

−12.15460830247379307927603948546, −10.90156753293625454957357693017, −10.18594489255706144591383469101, −9.053737875963196264335639400060, −8.206230609074418029123508264141, −7.21079631104616668703090107924, −5.61145107689088027019427940044, −4.54460232483215836114458519781, −2.87767099972560107222944647830, −1.82140145154670426448023246442, 0.22003390301990689060074640330, 3.03313153549556615385974929091, 4.71111621220657736462752399115, 5.24108362159591932745450376206, 6.79512380638695210153843715079, 7.51838152340578103169788170996, 8.626321482324035461232295396621, 9.656473103494246160283068476351, 10.30898219091986064067393496977, 11.36744345323551614103891956635

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