Properties

Label 2-16560-1.1-c1-0-24
Degree $2$
Conductor $16560$
Sign $1$
Analytic cond. $132.232$
Root an. cond. $11.4992$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 5·7-s + 4·13-s + 3·17-s + 4·19-s − 23-s + 25-s − 29-s − 31-s + 5·35-s − 37-s − 11·41-s − 4·43-s + 6·47-s + 18·49-s − 53-s − 59-s + 6·61-s + 4·65-s + 9·67-s + 13·71-s − 16·79-s + 9·83-s + 3·85-s − 8·89-s + 20·91-s + 4·95-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.88·7-s + 1.10·13-s + 0.727·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s − 0.185·29-s − 0.179·31-s + 0.845·35-s − 0.164·37-s − 1.71·41-s − 0.609·43-s + 0.875·47-s + 18/7·49-s − 0.137·53-s − 0.130·59-s + 0.768·61-s + 0.496·65-s + 1.09·67-s + 1.54·71-s − 1.80·79-s + 0.987·83-s + 0.325·85-s − 0.847·89-s + 2.09·91-s + 0.410·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16560\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(132.232\)
Root analytic conductor: \(11.4992\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 16560,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.919896633\)
\(L(\frac12)\) \(\approx\) \(3.919896633\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
23 \( 1 + T \)
good7 \( 1 - 5 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 - 13 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−15.82907108525449, −15.35357020497517, −14.70057473866886, −14.23641096277443, −13.79061159502061, −13.37493548454666, −12.50447780965310, −11.89611278071138, −11.39346233686850, −11.00452811643123, −10.30006348624295, −9.800208355116328, −8.932540890136984, −8.443104073852636, −7.988003760086389, −7.352450036402266, −6.658320094739152, −5.750342971558451, −5.333892197084605, −4.806433530168079, −3.928046259041324, −3.298683641499180, −2.205151663781755, −1.550795214607528, −0.9457395770098853, 0.9457395770098853, 1.550795214607528, 2.205151663781755, 3.298683641499180, 3.928046259041324, 4.806433530168079, 5.333892197084605, 5.750342971558451, 6.658320094739152, 7.352450036402266, 7.988003760086389, 8.443104073852636, 8.932540890136984, 9.800208355116328, 10.30006348624295, 11.00452811643123, 11.39346233686850, 11.89611278071138, 12.50447780965310, 13.37493548454666, 13.79061159502061, 14.23641096277443, 14.70057473866886, 15.35357020497517, 15.82907108525449

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