Properties

Label 2-87120-1.1-c1-0-120
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 + 3·7-s − 13-s − 7·17-s − 5·19-s − 8·23-s + 25-s − 5·29-s + 10·31-s + 3·35-s + 7·37-s + 6·41-s + 10·43-s − 10·47-s + 2·49-s + 4·53-s + 4·59-s − 65-s + 6·67-s + 9·71-s + 2·73-s + 6·79-s + 7·83-s − 7·85-s − 3·91-s − 5·95-s − 14·97-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.13·7-s − 0.277·13-s − 1.69·17-s − 1.14·19-s − 1.66·23-s + 1/5·25-s − 0.928·29-s + 1.79·31-s + 0.507·35-s + 1.15·37-s + 0.937·41-s + 1.52·43-s − 1.45·47-s + 2/7·49-s + 0.549·53-s + 0.520·59-s − 0.124·65-s + 0.733·67-s + 1.06·71-s + 0.234·73-s + 0.675·79-s + 0.768·83-s − 0.759·85-s − 0.314·91-s − 0.512·95-s − 1.42·97-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 \( 1 - 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−14.07854689260849, −13.71659316622823, −13.24553571198003, −12.63320362813543, −12.26964580193402, −11.48954071309933, −11.19825772091901, −10.81515909220734, −10.18417674137961, −9.616431541607559, −9.219805493375669, −8.463614878333041, −8.159183994412444, −7.750260470815406, −6.954427517803199, −6.354638237984945, −6.092900912793754, −5.307312048321161, −4.737939283881759, −4.195003960841689, −3.948789056450501, −2.643907209040748, −2.290023843994066, −1.843251727143892, −0.9354370444410662, 0, 0.9354370444410662, 1.843251727143892, 2.290023843994066, 2.643907209040748, 3.948789056450501, 4.195003960841689, 4.737939283881759, 5.307312048321161, 6.092900912793754, 6.354638237984945, 6.954427517803199, 7.750260470815406, 8.159183994412444, 8.463614878333041, 9.219805493375669, 9.616431541607559, 10.18417674137961, 10.81515909220734, 11.19825772091901, 11.48954071309933, 12.26964580193402, 12.63320362813543, 13.24553571198003, 13.71659316622823, 14.07854689260849

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