Properties

Label 2-2520-280.277-c0-0-3
Degree $2$
Conductor $2520$
Sign $0.999 + 0.0333i$
Analytic cond. $1.25764$
Root an. cond. $1.12144$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (0.965 − 0.258i)5-s + (0.5 − 0.866i)7-s + (0.707 + 0.707i)8-s + 10-s + (−1.67 − 0.965i)11-s + (0.707 − 0.707i)14-s + (0.500 + 0.866i)16-s + (0.965 + 0.258i)20-s + (−1.36 − 1.36i)22-s + (0.866 − 0.499i)25-s + (0.866 − 0.5i)28-s + 0.517·29-s + (−0.866 + 1.5i)31-s + (0.258 + 0.965i)32-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (0.965 − 0.258i)5-s + (0.5 − 0.866i)7-s + (0.707 + 0.707i)8-s + 10-s + (−1.67 − 0.965i)11-s + (0.707 − 0.707i)14-s + (0.500 + 0.866i)16-s + (0.965 + 0.258i)20-s + (−1.36 − 1.36i)22-s + (0.866 − 0.499i)25-s + (0.866 − 0.5i)28-s + 0.517·29-s + (−0.866 + 1.5i)31-s + (0.258 + 0.965i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.999 + 0.0333i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (1117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ 0.999 + 0.0333i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.551488575\)
\(L(\frac12)\) \(\approx\) \(2.551488575\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 \)
5 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good11 \( 1 + (1.67 + 0.965i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T^{2} \)
29 \( 1 - 0.517T + T^{2} \)
31 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.866 + 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (1.67 - 0.448i)T + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
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

−8.896779847237844755552111180080, −8.140328213295771209651050550866, −7.50756306209530348017447183920, −6.64145251167816093723430104378, −5.80278235551575740558083633432, −5.17022351319564278926726290428, −4.59051324897580181205242982269, −3.37168827378705085411580815639, −2.60731449690785535209206245488, −1.44081591806702413181888202336, 1.87353258294141004630001779394, 2.34034423307549558354033516478, 3.20731082282825408458241050309, 4.63418974721734456911770805032, 5.14054981906372861237618494669, 5.79303016096554265396449072370, 6.52844323394403133100358310198, 7.51378436384423260718728924490, 8.133270440955267772559678164863, 9.457189813153641840832465457328

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