↑

Properties

Label 2-2592-36.23-c1-0-23
Degree $2$
Conductor $2592$
Sign $0.984 - 0.173i$
Analytic cond. $20.6972$
Root an. cond. $4.54942$
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  + (2.25 + 1.30i)5-s + (−0.301 + 0.173i)7-s + (−0.336 − 0.582i)11-s + (−1.04 + 1.80i)13-s − 0.0255i·17-s − 5.29i·19-s + (4.44 − 7.70i)23-s + (0.890 + 1.54i)25-s + (0.133 − 0.0773i)29-s + (6.82 + 3.93i)31-s − 0.906·35-s + 5.34·37-s + (5.89 + 3.40i)41-s + (9.76 − 5.63i)43-s + (1.28 + 2.21i)47-s + ⋯
L(s)  = 1  + (1.00 + 0.582i)5-s + (−0.113 + 0.0657i)7-s + (−0.101 − 0.175i)11-s + (−0.288 + 0.499i)13-s − 0.00618i·17-s − 1.21i·19-s + (0.927 − 1.60i)23-s + (0.178 + 0.308i)25-s + (0.0248 − 0.0143i)29-s + (1.22 + 0.707i)31-s − 0.153·35-s + 0.879·37-s + (0.920 + 0.531i)41-s + (1.48 − 0.859i)43-s + (0.186 + 0.323i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(20.6972\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ 0.984 - 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.225524561\)
\(L(\frac12)\) \(\approx\) \(2.225524561\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.25 - 1.30i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.301 - 0.173i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.336 + 0.582i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.04 - 1.80i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.0255iT - 17T^{2} \)
19 \( 1 + 5.29iT - 19T^{2} \)
23 \( 1 + (-4.44 + 7.70i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.133 + 0.0773i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.82 - 3.93i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.34T + 37T^{2} \)
41 \( 1 + (-5.89 - 3.40i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-9.76 + 5.63i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.28 - 2.21i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.97iT - 53T^{2} \)
59 \( 1 + (3.07 - 5.32i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.26 - 10.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.80 + 2.77i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.82T + 71T^{2} \)
73 \( 1 - 7.21T + 73T^{2} \)
79 \( 1 + (-2.94 + 1.69i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.87 - 11.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 10.6iT - 89T^{2} \)
97 \( 1 + (-8.89 - 15.3i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.124666385947448118801245442365, −8.206927208515563130992402349611, −7.17702643748928066524537055055, −6.55438992549658765251647208192, −5.99339502373102696127136093284, −4.95868764800503704216527140252, −4.25364197268249073011854670697, −2.69390171514221935802327780877, −2.54919038963929140927514846605, −0.955775549505788774443930154474, 0.994684830455693419435629761638, 2.00415024261827113081434567468, 3.04883864543179203020978391003, 4.12056268939072727922434625696, 5.11448609561189586942884284987, 5.71797031471213633798610268627, 6.35900123976726877915226030662, 7.51657545350750767881367660769, 7.993609769787454772023481548487, 9.066207153444272368357285490583

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