↑

Properties

Label 2-1080-40.29-c1-0-89
Degree $2$
Conductor $1080$
Sign $-0.842 - 0.538i$
Analytic cond. $8.62384$
Root an. cond. $2.93663$
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  + (−0.986 − 1.01i)2-s + (−0.0536 + 1.99i)4-s + (−0.569 − 2.16i)5-s − 4.64i·7-s + (2.07 − 1.91i)8-s + (−1.62 + 2.71i)10-s − 3.44i·11-s − 2.72·13-s + (−4.70 + 4.58i)14-s + (−3.99 − 0.214i)16-s − 2.43i·17-s + 7.45i·19-s + (4.35 − 1.02i)20-s + (−3.49 + 3.40i)22-s − 6.60i·23-s + ⋯
L(s)  = 1  + (−0.697 − 0.716i)2-s + (−0.0268 + 0.999i)4-s + (−0.254 − 0.967i)5-s − 1.75i·7-s + (0.734 − 0.678i)8-s + (−0.515 + 0.857i)10-s − 1.04i·11-s − 0.756·13-s + (−1.25 + 1.22i)14-s + (−0.998 − 0.0536i)16-s − 0.589i·17-s + 1.71i·19-s + (0.973 − 0.228i)20-s + (−0.745 + 0.725i)22-s − 1.37i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.842 - 0.538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.842 - 0.538i$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ -0.842 - 0.538i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6794208032\)
\(L(\frac12)\) \(\approx\) \(0.6794208032\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.986 + 1.01i)T \)
3 \( 1 \)
5 \( 1 + (0.569 + 2.16i)T \)
good7 \( 1 + 4.64iT - 7T^{2} \)
11 \( 1 + 3.44iT - 11T^{2} \)
13 \( 1 + 2.72T + 13T^{2} \)
17 \( 1 + 2.43iT - 17T^{2} \)
19 \( 1 - 7.45iT - 19T^{2} \)
23 \( 1 + 6.60iT - 23T^{2} \)
29 \( 1 + 0.952iT - 29T^{2} \)
31 \( 1 - 5.77T + 31T^{2} \)
37 \( 1 + 0.0145T + 37T^{2} \)
41 \( 1 - 4.42T + 41T^{2} \)
43 \( 1 - 6.83T + 43T^{2} \)
47 \( 1 - 6.05iT - 47T^{2} \)
53 \( 1 + 4.54T + 53T^{2} \)
59 \( 1 - 9.70iT - 59T^{2} \)
61 \( 1 + 1.57iT - 61T^{2} \)
67 \( 1 + 7.29T + 67T^{2} \)
71 \( 1 + 12.8T + 71T^{2} \)
73 \( 1 + 3.38iT - 73T^{2} \)
79 \( 1 - 5.03T + 79T^{2} \)
83 \( 1 + 12.7T + 83T^{2} \)
89 \( 1 - 0.535T + 89T^{2} \)
97 \( 1 + 9.22iT - 97T^{2} \)
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

−9.525390819778078922599219790123, −8.557132309242546801361678672007, −7.87211764996745110475372265061, −7.31364447705037642291976421721, −6.07555053817594098562258100958, −4.55262458740362395778493073842, −4.09121443093621793375306009714, −2.95610303695811000719485608067, −1.30057998627819342659795186228, −0.40871048891713804537776766710, 2.03360786718719284427527110048, 2.80639440827526143088861721521, 4.57994580472036134940055903601, 5.45875310722083212651033075145, 6.30933400031956096080199966975, 7.11341242486332411091113382353, 7.77582417613463144023701391937, 8.781568089798085685364330430422, 9.451424784413737415134836916105, 10.05894781639472675029528692966

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