↑

Properties

Label 2-2592-9.7-c1-0-27
Degree $2$
Conductor $2592$
Sign $-0.173 + 0.984i$
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  + (−1 − 1.73i)5-s + (−1.5 + 2.59i)7-s + (−3 + 5.19i)11-s + (1.5 + 2.59i)13-s − 2·17-s − 3·19-s + (−3 − 5.19i)23-s + (0.500 − 0.866i)25-s + (4 − 6.92i)29-s + 6·35-s + 7·37-s + (−4 − 6.92i)41-s + (6 − 10.3i)43-s + (−3 + 5.19i)47-s + (−1 − 1.73i)49-s + ⋯
L(s)  = 1  + (−0.447 − 0.774i)5-s + (−0.566 + 0.981i)7-s + (−0.904 + 1.56i)11-s + (0.416 + 0.720i)13-s − 0.485·17-s − 0.688·19-s + (−0.625 − 1.08i)23-s + (0.100 − 0.173i)25-s + (0.742 − 1.28i)29-s + 1.01·35-s + 1.15·37-s + (−0.624 − 1.08i)41-s + (0.914 − 1.58i)43-s + (−0.437 + 0.757i)47-s + (−0.142 − 0.247i)49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(20.6972\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6200860753\)
\(L(\frac12)\) \(\approx\) \(0.6200860753\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.5 - 2.59i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 3T + 19T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4 + 6.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7T + 37T^{2} \)
41 \( 1 + (4 + 6.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6 + 10.3i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 15T + 73T^{2} \)
79 \( 1 + (4.5 - 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + (4.5 - 7.79i)T + (-48.5 - 84.0i)T^{2} \)
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

−8.601966968034222253608195119857, −8.131653467419613408023727006837, −7.11023939657461467003850127107, −6.36515226335793498346682807937, −5.54893302847208674734349387869, −4.46231438940450273872916865404, −4.24449646441145338824423210098, −2.59741191012505816484055479132, −2.04978175212343463560244884481, −0.23429718909680251568145510846, 1.01838773833009563796825520177, 2.82073206755928660023093256695, 3.30885124232416514397037093332, 4.10052589065827996560663852422, 5.28163119026811446528389108271, 6.16043664778174909414480651459, 6.74721077846424217419143685184, 7.69139172072118987836415953332, 8.126885370068648381292660471126, 9.011996121292348459824753466404

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