↑

Properties

Label 2-24e2-144.85-c1-0-6
Degree $2$
Conductor $576$
Sign $-0.598 - 0.800i$
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  + (1.64 + 0.543i)3-s + (−1.60 + 0.430i)5-s + (−3.62 + 2.09i)7-s + (2.40 + 1.78i)9-s + (−1.24 + 4.63i)11-s + (−0.879 − 3.28i)13-s + (−2.87 − 0.164i)15-s − 2.14·17-s + (−1.03 + 1.03i)19-s + (−7.10 + 1.47i)21-s + (0.405 + 0.234i)23-s + (−1.93 + 1.11i)25-s + (2.99 + 4.24i)27-s + (6.55 + 1.75i)29-s + (−3.18 + 5.50i)31-s + ⋯
L(s)  = 1  + (0.949 + 0.313i)3-s + (−0.718 + 0.192i)5-s + (−1.37 + 0.791i)7-s + (0.803 + 0.595i)9-s + (−0.374 + 1.39i)11-s + (−0.243 − 0.910i)13-s + (−0.742 − 0.0425i)15-s − 0.519·17-s + (−0.236 + 0.236i)19-s + (−1.55 + 0.321i)21-s + (0.0846 + 0.0488i)23-s + (−0.387 + 0.223i)25-s + (0.575 + 0.817i)27-s + (1.21 + 0.326i)29-s + (−0.571 + 0.989i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.598 - 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.598 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.598 - 0.800i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.598 - 0.800i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.495952 + 0.990046i\)
\(L(\frac12)\) \(\approx\) \(0.495952 + 0.990046i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 - 0.543i)T \)
good5 \( 1 + (1.60 - 0.430i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (3.62 - 2.09i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.24 - 4.63i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.879 + 3.28i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 2.14T + 17T^{2} \)
19 \( 1 + (1.03 - 1.03i)T - 19iT^{2} \)
23 \( 1 + (-0.405 - 0.234i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.55 - 1.75i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (3.18 - 5.50i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.728 + 0.728i)T + 37iT^{2} \)
41 \( 1 + (-2.52 - 1.45i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.84 + 10.6i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-4.61 - 7.99i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.17 - 1.17i)T + 53iT^{2} \)
59 \( 1 + (-1.48 + 0.397i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-7.53 - 2.01i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.63 + 9.82i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 8.27iT - 71T^{2} \)
73 \( 1 - 8.16iT - 73T^{2} \)
79 \( 1 + (3.63 + 6.28i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.59 - 2.03i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 11.1iT - 89T^{2} \)
97 \( 1 + (5.67 + 9.83i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.72410076131005357433449251442, −10.03645977799298268169169004482, −9.307738967947634248182739425544, −8.474176464382186023869646238626, −7.49088920107168137068023452028, −6.80256082925134776897145906187, −5.40097813845436732935521053802, −4.21994678044034961907258922823, −3.19317868688886526340960818078, −2.34977185738555726463372064978, 0.53589971476854808726480518849, 2.59621183944229669821877583172, 3.62122593449102767490861177722, 4.32039315076205630692727144572, 6.16225572279122180456375329738, 6.89718420704194700191153765355, 7.80070748143994322087542688356, 8.635782648501544674402397146129, 9.411896401542017971262464495886, 10.27475823551193666596254787901

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