Properties

Label 2-87120-1.1-c1-0-12
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s − 3·13-s − 17-s − 5·19-s + 4·23-s + 25-s − 9·29-s − 6·31-s + 35-s − 11·37-s − 2·41-s + 2·43-s − 6·47-s − 6·49-s − 4·53-s + 4·59-s − 4·61-s − 3·65-s + 6·67-s + 5·71-s − 6·73-s − 14·79-s − 7·83-s − 85-s − 4·89-s − 3·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 0.832·13-s − 0.242·17-s − 1.14·19-s + 0.834·23-s + 1/5·25-s − 1.67·29-s − 1.07·31-s + 0.169·35-s − 1.80·37-s − 0.312·41-s + 0.304·43-s − 0.875·47-s − 6/7·49-s − 0.549·53-s + 0.520·59-s − 0.512·61-s − 0.372·65-s + 0.733·67-s + 0.593·71-s − 0.702·73-s − 1.57·79-s − 0.768·83-s − 0.108·85-s − 0.423·89-s − 0.314·91-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: \(0\)
Selberg data: \((2,\ 87120,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9221179774\)
\(L(\frac12)\) \(\approx\) \(0.9221179774\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.07998802698691, −13.23101560012052, −12.90417063087869, −12.62475411890534, −11.93478249229179, −11.24366315103163, −11.05603319853242, −10.44524388519284, −9.871772890976087, −9.460969996241278, −8.782132481494407, −8.559288803613256, −7.781251100950887, −7.188234047073862, −6.899917125977610, −6.193701467079902, −5.562322538657568, −5.111403189704185, −4.622764559899624, −3.907398526466297, −3.308399182786401, −2.591977869629392, −1.822376095786657, −1.608741223022018, −0.2856726985025513, 0.2856726985025513, 1.608741223022018, 1.822376095786657, 2.591977869629392, 3.308399182786401, 3.907398526466297, 4.622764559899624, 5.111403189704185, 5.562322538657568, 6.193701467079902, 6.899917125977610, 7.188234047073862, 7.781251100950887, 8.559288803613256, 8.782132481494407, 9.460969996241278, 9.871772890976087, 10.44524388519284, 11.05603319853242, 11.24366315103163, 11.93478249229179, 12.62475411890534, 12.90417063087869, 13.23101560012052, 14.07998802698691

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