Properties

Label 2-92736-1.1-c1-0-71
Degree $2$
Conductor $92736$
Sign $-1$
Analytic cond. $740.500$
Root an. cond. $27.2121$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 7-s − 4·11-s − 4·13-s + 8·17-s + 2·19-s − 23-s − 25-s + 2·29-s − 6·31-s − 2·35-s + 10·37-s − 6·41-s + 8·43-s − 6·47-s + 49-s + 2·53-s + 8·55-s − 10·61-s + 8·65-s − 8·67-s + 12·71-s + 6·73-s − 4·77-s + 2·83-s − 16·85-s − 12·89-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s − 1.20·11-s − 1.10·13-s + 1.94·17-s + 0.458·19-s − 0.208·23-s − 1/5·25-s + 0.371·29-s − 1.07·31-s − 0.338·35-s + 1.64·37-s − 0.937·41-s + 1.21·43-s − 0.875·47-s + 1/7·49-s + 0.274·53-s + 1.07·55-s − 1.28·61-s + 0.992·65-s − 0.977·67-s + 1.42·71-s + 0.702·73-s − 0.455·77-s + 0.219·83-s − 1.73·85-s − 1.27·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92736\)    =    \(2^{6} \cdot 3^{2} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(740.500\)
Root analytic conductor: \(27.2121\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{92736} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 92736,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - T \)
23 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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

−14.07044347734954, −13.71366704567893, −12.93242889338174, −12.52278932150370, −12.15054815303238, −11.67781000408589, −11.20247212792616, −10.59319431584490, −10.16295894269922, −9.626939846166066, −9.242623099395244, −8.299110729660464, −7.876297087696069, −7.678595313427348, −7.297362138039080, −6.494841052035425, −5.622399099797511, −5.414017263652105, −4.791403918533788, −4.226135253872640, −3.538122632020381, −2.988784255117605, −2.454399854474897, −1.604726984896727, −0.7604112020528307, 0, 0.7604112020528307, 1.604726984896727, 2.454399854474897, 2.988784255117605, 3.538122632020381, 4.226135253872640, 4.791403918533788, 5.414017263652105, 5.622399099797511, 6.494841052035425, 7.297362138039080, 7.678595313427348, 7.876297087696069, 8.299110729660464, 9.242623099395244, 9.626939846166066, 10.16295894269922, 10.59319431584490, 11.20247212792616, 11.67781000408589, 12.15054815303238, 12.52278932150370, 12.93242889338174, 13.71366704567893, 14.07044347734954

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