↑

Properties

Label 2-2592-72.11-c1-0-25
Degree $2$
Conductor $2592$
Sign $0.787 - 0.616i$
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  + (0.893 + 1.54i)5-s + (2.90 + 1.67i)7-s + (3.13 + 1.81i)11-s + (−1.48 + 0.859i)13-s − 6.25i·17-s + 4.57·19-s + (0.308 + 0.534i)23-s + (0.902 − 1.56i)25-s + (2.83 − 4.91i)29-s + (7.01 − 4.04i)31-s + 6.00i·35-s + 7.06i·37-s + (5.99 − 3.46i)41-s + (−3.33 + 5.77i)43-s + (0.506 − 0.876i)47-s + ⋯
L(s)  = 1  + (0.399 + 0.692i)5-s + (1.09 + 0.634i)7-s + (0.946 + 0.546i)11-s + (−0.412 + 0.238i)13-s − 1.51i·17-s + 1.04·19-s + (0.0643 + 0.111i)23-s + (0.180 − 0.312i)25-s + (0.526 − 0.912i)29-s + (1.25 − 0.726i)31-s + 1.01i·35-s + 1.16i·37-s + (0.936 − 0.540i)41-s + (−0.508 + 0.880i)43-s + (0.0738 − 0.127i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.787 - 0.616i$
Analytic conductor: \(20.6972\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ 0.787 - 0.616i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.522336137\)
\(L(\frac12)\) \(\approx\) \(2.522336137\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.893 - 1.54i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.90 - 1.67i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.13 - 1.81i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.48 - 0.859i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.25iT - 17T^{2} \)
19 \( 1 - 4.57T + 19T^{2} \)
23 \( 1 + (-0.308 - 0.534i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.83 + 4.91i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.01 + 4.04i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.06iT - 37T^{2} \)
41 \( 1 + (-5.99 + 3.46i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.33 - 5.77i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.506 + 0.876i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 10.5T + 53T^{2} \)
59 \( 1 + (4.33 - 2.50i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-12.3 - 7.14i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.69 + 8.12i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.70T + 71T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 + (-13.3 - 7.71i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.60 + 3.23i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 1.45iT - 89T^{2} \)
97 \( 1 + (-2.92 + 5.07i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.095733518763858505696463074617, −8.118245115567917874747059769662, −7.43772706749664603121615475415, −6.67610716761368175045881897769, −5.93947416088256785582959553958, −4.89981572531585443140085229878, −4.44996693960455637709860725314, −2.98411665904827537819318669540, −2.34058316175048017656510063063, −1.18968727598357613874741524059, 1.07451623849735622676047389891, 1.62538577482141716619329984204, 3.14289552554765495884156161764, 4.12567237029166381970521067362, 4.86275435812185726722120189207, 5.59429585529344395796573040909, 6.46845457125241792822327965288, 7.36913901844289337194709589876, 8.158652452407325139233560627178, 8.722340226215622484390650031562

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