↑

Properties

Label 2-264-88.21-c2-0-43
Degree $2$
Conductor $264$
Sign $0.159 + 0.987i$
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  + (1.20 + 1.59i)2-s − 1.73i·3-s + (−1.09 + 3.84i)4-s − 3.79i·5-s + (2.76 − 2.08i)6-s − 12.8i·7-s + (−7.46 + 2.88i)8-s − 2.99·9-s + (6.05 − 4.56i)10-s + (−10.7 − 2.27i)11-s + (6.66 + 1.89i)12-s − 21.4·13-s + (20.5 − 15.5i)14-s − 6.56·15-s + (−13.5 − 8.43i)16-s − 2.60i·17-s + ⋯
L(s)  = 1  + (0.602 + 0.798i)2-s − 0.577i·3-s + (−0.273 + 0.961i)4-s − 0.758i·5-s + (0.460 − 0.347i)6-s − 1.83i·7-s + (−0.932 + 0.360i)8-s − 0.333·9-s + (0.605 − 0.456i)10-s + (−0.978 − 0.207i)11-s + (0.555 + 0.158i)12-s − 1.65·13-s + (1.46 − 1.10i)14-s − 0.437·15-s + (−0.849 − 0.527i)16-s − 0.153i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $0.159 + 0.987i$
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: \((2,\ 264,\ (\ :1),\ 0.159 + 0.987i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.09058 - 0.928358i\)
\(L(\frac12)\) \(\approx\) \(1.09058 - 0.928358i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.20 - 1.59i)T \)
3 \( 1 + 1.73iT \)
11 \( 1 + (10.7 + 2.27i)T \)
good5 \( 1 + 3.79iT - 25T^{2} \)
7 \( 1 + 12.8iT - 49T^{2} \)
13 \( 1 + 21.4T + 169T^{2} \)
17 \( 1 + 2.60iT - 289T^{2} \)
19 \( 1 - 17.4T + 361T^{2} \)
23 \( 1 - 14.5T + 529T^{2} \)
29 \( 1 - 23.4T + 841T^{2} \)
31 \( 1 - 35.3T + 961T^{2} \)
37 \( 1 + 40.6iT - 1.36e3T^{2} \)
41 \( 1 - 26.8iT - 1.68e3T^{2} \)
43 \( 1 + 36.4T + 1.84e3T^{2} \)
47 \( 1 - 28.7T + 2.20e3T^{2} \)
53 \( 1 + 14.2iT - 2.80e3T^{2} \)
59 \( 1 - 46.1iT - 3.48e3T^{2} \)
61 \( 1 + 39.1T + 3.72e3T^{2} \)
67 \( 1 + 9.26iT - 4.48e3T^{2} \)
71 \( 1 - 124.T + 5.04e3T^{2} \)
73 \( 1 + 91.6iT - 5.32e3T^{2} \)
79 \( 1 + 111. iT - 6.24e3T^{2} \)
83 \( 1 - 24.4T + 6.88e3T^{2} \)
89 \( 1 - 46.7T + 7.92e3T^{2} \)
97 \( 1 + 158.T + 9.40e3T^{2} \)
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

−11.92397726882662946589746324199, −10.60452717250286116333048968086, −9.490932982698448429576305697401, −8.101924598134276969753405711637, −7.48509409520799918562494746499, −6.76421660592781456072913488291, −5.19207857941437315354738091161, −4.54111713004399960196999650555, −2.97683140689805348978472563434, −0.58245858337864352568443611230, 2.52030904721010699150909522035, 2.93729944066176310186666340984, 4.89202319263961892075350469382, 5.39296689078214471888963180811, 6.70825396071741491543582328177, 8.323280860151306114720649308412, 9.519083862420291202997116464012, 10.06798170957448032219081070060, 11.10065866995451271992172203585, 12.04786779922182112589390090941

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