↑

Properties

Label 2-72e2-8.5-c1-0-75
Degree $2$
Conductor $5184$
Sign $-0.258 + 0.965i$
Analytic cond. $41.3944$
Root an. cond. $6.43385$
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  + 3.46i·5-s − 3i·11-s − 3.46i·13-s + 3·17-s + 7i·19-s − 3.46·23-s − 6.99·25-s − 6.92i·29-s − 6.92·31-s − 10.3i·37-s − 3·41-s + 5i·43-s − 3.46·47-s − 7·49-s − 13.8i·53-s + ⋯
L(s)  = 1  + 1.54i·5-s − 0.904i·11-s − 0.960i·13-s + 0.727·17-s + 1.60i·19-s − 0.722·23-s − 1.39·25-s − 1.28i·29-s − 1.24·31-s − 1.70i·37-s − 0.468·41-s + 0.762i·43-s − 0.505·47-s − 49-s − 1.90i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $-0.258 + 0.965i$
Analytic conductor: \(41.3944\)
Root analytic conductor: \(6.43385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5184} (2593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5184,\ (\ :1/2),\ -0.258 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6063052696\)
\(L(\frac12)\) \(\approx\) \(0.6063052696\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.46iT - 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 3iT - 11T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - 7iT - 19T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 + 6.92iT - 29T^{2} \)
31 \( 1 + 6.92T + 31T^{2} \)
37 \( 1 + 10.3iT - 37T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - 5iT - 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 + 13.8iT - 53T^{2} \)
59 \( 1 + 9iT - 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 - 5iT - 67T^{2} \)
71 \( 1 - 3.46T + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 + 17.3T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - T + 97T^{2} \)
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

−7.913578928086096501148644980707, −7.40292273558658904960746319097, −6.46829534214700945962338046995, −5.87191459806522663223959825102, −5.42132526625446890216272544740, −3.87966230575842623350765032061, −3.50292368981190996222767589710, −2.72127008149144278018355967896, −1.73250735540054602263648648321, −0.15856651861005386993742275142, 1.24087962143846434947256731216, 1.89833769763077739653451697571, 3.14791377463458503309007706653, 4.23407018868307354391966125510, 4.78947827808530031307629042212, 5.25202730077320358944092362033, 6.25744943957183932847864289414, 7.09988582065497167425131429381, 7.67052299236597057984919401326, 8.682578307334547244748398175111

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