Properties

Label 2-3072-8.5-c1-0-0
Degree $2$
Conductor $3072$
Sign $i$
Analytic cond. $24.5300$
Root an. cond. $4.95278$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + 3.86i·5-s − 2.44·7-s − 9-s + 1.46i·11-s − 4.24i·13-s − 3.86·15-s − 3.46·17-s + 7.46i·19-s − 2.44i·21-s − 2.82·23-s − 9.92·25-s i·27-s − 8.76i·29-s + 7.34·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.72i·5-s − 0.925·7-s − 0.333·9-s + 0.441i·11-s − 1.17i·13-s − 0.997·15-s − 0.840·17-s + 1.71i·19-s − 0.534i·21-s − 0.589·23-s − 1.98·25-s − 0.192i·27-s − 1.62i·29-s + 1.31·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3072\)    =    \(2^{10} \cdot 3\)
Sign: $i$
Analytic conductor: \(24.5300\)
Root analytic conductor: \(4.95278\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3072} (1537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3072,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.09116190786\)
\(L(\frac12)\) \(\approx\) \(0.09116190786\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - iT \)
good5 \( 1 - 3.86iT - 5T^{2} \)
7 \( 1 + 2.44T + 7T^{2} \)
11 \( 1 - 1.46iT - 11T^{2} \)
13 \( 1 + 4.24iT - 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 - 7.46iT - 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 + 8.76iT - 29T^{2} \)
31 \( 1 - 7.34T + 31T^{2} \)
37 \( 1 - 0.656iT - 37T^{2} \)
41 \( 1 + 4.53T + 41T^{2} \)
43 \( 1 - 3.46iT - 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + 4.62iT - 53T^{2} \)
59 \( 1 + 13.8iT - 59T^{2} \)
61 \( 1 - 0.656iT - 61T^{2} \)
67 \( 1 - 14.9iT - 67T^{2} \)
71 \( 1 + 6.41T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + 2.44T + 79T^{2} \)
83 \( 1 - 5.46iT - 83T^{2} \)
89 \( 1 - 4.92T + 89T^{2} \)
97 \( 1 - 1.07T + 97T^{2} \)
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

−9.755855495988990076280382387260, −8.331085092823983209661119193547, −7.82972179673772547113292579769, −6.86755998564392336565790160754, −6.23406130714241212715902011430, −5.75944391918282797710163914057, −4.42389969232881210347065700722, −3.57050508329634882885912643450, −3.00525924559756471847679565562, −2.15104830739903089818967318878, 0.03022247252965957023753620645, 1.09096439818660904477654821370, 2.14604808605590729457305085554, 3.28606272789826592172688168510, 4.47008685706438419738489379979, 4.88669786150696648805137585280, 5.96795692172599914990561958948, 6.62246957677210623859672863490, 7.30920382979036547294721965704, 8.408453885976097740598271790845

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