↑

Properties

Label 2-2592-72.61-c1-0-19
Degree $2$
Conductor $2592$
Sign $0.906 + 0.422i$
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  + (−3.23 + 1.86i)5-s + (−0.366 + 0.633i)7-s + (−4.09 − 2.36i)11-s + (−2.13 + 1.23i)13-s − 3.73·17-s − 3.26i·19-s + (4.36 + 7.56i)23-s + (4.46 − 7.73i)25-s + (−4.5 − 2.59i)29-s + (1 + 1.73i)31-s − 2.73i·35-s − 5i·37-s + (−2.26 − 3.92i)41-s + (8.36 + 4.83i)43-s + (1.73 − 3i)47-s + ⋯
L(s)  = 1  + (−1.44 + 0.834i)5-s + (−0.138 + 0.239i)7-s + (−1.23 − 0.713i)11-s + (−0.591 + 0.341i)13-s − 0.905·17-s − 0.749i·19-s + (0.910 + 1.57i)23-s + (0.892 − 1.54i)25-s + (−0.835 − 0.482i)29-s + (0.179 + 0.311i)31-s − 0.461i·35-s − 0.821i·37-s + (−0.354 − 0.613i)41-s + (1.27 + 0.736i)43-s + (0.252 − 0.437i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.906 + 0.422i$
Analytic conductor: \(20.6972\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (2161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ 0.906 + 0.422i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6493362866\)
\(L(\frac12)\) \(\approx\) \(0.6493362866\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.23 - 1.86i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.366 - 0.633i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.09 + 2.36i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.13 - 1.23i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.73T + 17T^{2} \)
19 \( 1 + 3.26iT - 19T^{2} \)
23 \( 1 + (-4.36 - 7.56i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.5 + 2.59i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 5iT - 37T^{2} \)
41 \( 1 + (2.26 + 3.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.36 - 4.83i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.73 + 3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.928iT - 53T^{2} \)
59 \( 1 + (-7.26 + 4.19i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.79 + 4.5i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.09 - 2.36i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.66T + 71T^{2} \)
73 \( 1 - 9T + 73T^{2} \)
79 \( 1 + (-2.63 + 4.56i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-12.1 - 7i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 11.7T + 89T^{2} \)
97 \( 1 + (-4 + 6.92i)T + (-48.5 - 84.0i)T^{2} \)
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

−8.818048348591536578466778969070, −7.79998774068367156387969149496, −7.45900808866195960902854001164, −6.75299546248311619799005159807, −5.69692089716511367365678360826, −4.84201474195858837567206248105, −3.90399000499255491286348398014, −3.09439792911219591089330610541, −2.36699045680888375322315118691, −0.35429115146978957963573755423, 0.65560566286579133030351953302, 2.26872601464439317032929646687, 3.30308329041394444165461313003, 4.42485038872002155908455430484, 4.69944068835344600994320693818, 5.66597926143285930294170202492, 6.97029967071046909595713918877, 7.44469447095679338129043963512, 8.200507525608738759717967143756, 8.695224977703199624047815579981

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