Properties

Label 2-24e2-12.11-c1-0-1
Degree $2$
Conductor $576$
Sign $-0.577 - 0.816i$
Analytic cond. $4.59938$
Root an. cond. $2.14461$
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  + 4.24i·5-s − 4·13-s + 4.24i·17-s − 12.9·25-s − 4.24i·29-s − 2·37-s + 12.7i·41-s + 7·49-s + 12.7i·53-s + 10·61-s − 16.9i·65-s + 16·73-s − 17.9·85-s − 4.24i·89-s − 8·97-s + ⋯
L(s)  = 1  + 1.89i·5-s − 1.10·13-s + 1.02i·17-s − 2.59·25-s − 0.787i·29-s − 0.328·37-s + 1.98i·41-s + 49-s + 1.74i·53-s + 1.28·61-s − 2.10i·65-s + 1.87·73-s − 1.95·85-s − 0.449i·89-s − 0.812·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.510104 + 0.985446i\)
\(L(\frac12)\) \(\approx\) \(0.510104 + 0.985446i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 - 4.24iT - 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 - 4.24iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 4.24iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 12.7iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 12.7iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 16T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 4.24iT - 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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

−10.94218424541266876694693040738, −10.20922980389346248944772950301, −9.600632917321240083368487812174, −8.155272001130774117625398282381, −7.36335648202768566384320099726, −6.59331539728677042253254300724, −5.77151805299338591681195647618, −4.27094011215157672317506564696, −3.13802009461373358003284695538, −2.21567929648355828327757462051, 0.61016965810068078494371946019, 2.14747593335326535874907222751, 3.88228561683618643959513697057, 5.04012730347137370705953888353, 5.34784401701713482199570853103, 6.93287163823627938508719895635, 7.87570620009373905343372266629, 8.806814450412389348985876620774, 9.364972501136662661314388121361, 10.21556876712294627339832575986

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