Properties

Label 2-2592-36.11-c1-0-19
Degree $2$
Conductor $2592$
Sign $0.819 - 0.573i$
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  + (0.896 − 0.517i)5-s + (−0.232 − 0.133i)7-s + (1.93 − 3.34i)11-s + (1.23 + 2.13i)13-s + 6.69i·17-s + 1.73i·19-s + (2.96 + 5.13i)23-s + (−1.96 + 3.40i)25-s + (−1.79 − 1.03i)29-s + (−0.464 + 0.267i)31-s − 0.277·35-s − 6.46·37-s + (1.79 − 1.03i)41-s + (6.46 + 3.73i)43-s + (4.76 − 8.24i)47-s + ⋯
L(s)  = 1  + (0.400 − 0.231i)5-s + (−0.0877 − 0.0506i)7-s + (0.582 − 1.00i)11-s + (0.341 + 0.591i)13-s + 1.62i·17-s + 0.397i·19-s + (0.618 + 1.07i)23-s + (−0.392 + 0.680i)25-s + (−0.332 − 0.192i)29-s + (−0.0833 + 0.0481i)31-s − 0.0468·35-s − 1.06·37-s + (0.280 − 0.161i)41-s + (0.985 + 0.569i)43-s + (0.694 − 1.20i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.819 - 0.573i$
Analytic conductor: \(20.6972\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ 0.819 - 0.573i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.948466494\)
\(L(\frac12)\) \(\approx\) \(1.948466494\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.896 + 0.517i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.232 + 0.133i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.93 + 3.34i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.23 - 2.13i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 6.69iT - 17T^{2} \)
19 \( 1 - 1.73iT - 19T^{2} \)
23 \( 1 + (-2.96 - 5.13i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.79 + 1.03i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.464 - 0.267i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.46T + 37T^{2} \)
41 \( 1 + (-1.79 + 1.03i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.46 - 3.73i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.76 + 8.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 13.3iT - 53T^{2} \)
59 \( 1 + (-3.72 - 6.45i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.69 + 8.13i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.42 + 4.86i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 - 9.92T + 73T^{2} \)
79 \( 1 + (13.1 + 7.59i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.86 + 6.69i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 6.69iT - 89T^{2} \)
97 \( 1 + (3.5 - 6.06i)T + (-48.5 - 84.0i)T^{2} \)
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

−8.976216845624025398167031913455, −8.331813131765413765025467753171, −7.45709461740305352213748285212, −6.50239536224679545178995704910, −5.89255746272879178862122922880, −5.22885696574185910722206994433, −3.86996242198180309206746595589, −3.55962063710939476182441538773, −2.03438161207632967373953519276, −1.16496040681687428794552693629, 0.72919316802352239534181730544, 2.15171726885582872712061999835, 2.90798847464328010603789018021, 4.05966188403011215611478737935, 4.90855195435289578390723795874, 5.65641942812137175921893282084, 6.74631975154939723258969673776, 7.03907610673200488440964689177, 8.052860147589780495126459341887, 8.923447831706242617852176321119

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