Properties

Label 2-87120-1.1-c1-0-109
Degree $2$
Conductor $87120$
Sign $-1$
Analytic cond. $695.656$
Root an. cond. $26.3753$
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  + 5-s − 2·13-s − 6·17-s − 2·23-s + 25-s + 8·29-s − 4·31-s − 2·37-s + 4·41-s + 4·43-s + 2·47-s − 7·49-s + 10·53-s + 8·61-s − 2·65-s + 2·67-s + 8·71-s − 10·73-s + 4·79-s + 12·83-s − 6·85-s − 6·89-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.554·13-s − 1.45·17-s − 0.417·23-s + 1/5·25-s + 1.48·29-s − 0.718·31-s − 0.328·37-s + 0.624·41-s + 0.609·43-s + 0.291·47-s − 49-s + 1.37·53-s + 1.02·61-s − 0.248·65-s + 0.244·67-s + 0.949·71-s − 1.17·73-s + 0.450·79-s + 1.31·83-s − 0.650·85-s − 0.635·89-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87120\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(695.656\)
Root analytic conductor: \(26.3753\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{87120} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 87120,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 - T \)
11 \( 1 \)
good7 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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

−14.15553779156746, −13.61619767707649, −13.21344814302924, −12.71865389693240, −12.15586152424549, −11.77736358777178, −11.06824704626415, −10.69515420466311, −10.22049831315940, −9.555471988301098, −9.273129851505812, −8.566083293729621, −8.255700056665394, −7.492519575210188, −6.952118195871594, −6.527218356128202, −5.995484793664374, −5.293408341351496, −4.872426596144867, −4.192683720124707, −3.744977223903908, −2.723774827639860, −2.437853117932002, −1.741613974215868, −0.8758171006602686, 0, 0.8758171006602686, 1.741613974215868, 2.437853117932002, 2.723774827639860, 3.744977223903908, 4.192683720124707, 4.872426596144867, 5.293408341351496, 5.995484793664374, 6.527218356128202, 6.952118195871594, 7.492519575210188, 8.255700056665394, 8.566083293729621, 9.273129851505812, 9.555471988301098, 10.22049831315940, 10.69515420466311, 11.06824704626415, 11.77736358777178, 12.15586152424549, 12.71865389693240, 13.21344814302924, 13.61619767707649, 14.15553779156746

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