Properties

Label 2-87120-1.1-c1-0-128
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 2·7-s − 2·13-s − 6·17-s − 4·19-s + 6·23-s + 25-s + 6·29-s + 4·31-s + 2·35-s + 2·37-s + 6·41-s − 10·43-s − 6·47-s − 3·49-s + 6·53-s + 12·59-s − 2·61-s − 2·65-s − 2·67-s − 12·71-s − 2·73-s + 8·79-s − 6·83-s − 6·85-s + 6·89-s − 4·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.755·7-s − 0.554·13-s − 1.45·17-s − 0.917·19-s + 1.25·23-s + 1/5·25-s + 1.11·29-s + 0.718·31-s + 0.338·35-s + 0.328·37-s + 0.937·41-s − 1.52·43-s − 0.875·47-s − 3/7·49-s + 0.824·53-s + 1.56·59-s − 0.256·61-s − 0.248·65-s − 0.244·67-s − 1.42·71-s − 0.234·73-s + 0.900·79-s − 0.658·83-s − 0.650·85-s + 0.635·89-s − 0.419·91-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: \(1\)
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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.27341448483639, −13.47773098737986, −13.23575282022768, −12.84005555499916, −12.09001444702613, −11.67420235228204, −11.11699106854549, −10.77520789255849, −10.14048903183742, −9.763377654587806, −9.018108431516341, −8.589617607733836, −8.304175967539516, −7.523248667242851, −6.949381606670196, −6.512456219741193, −6.050894102451226, −5.187527986387865, −4.741381990358685, −4.481563666123548, −3.647405496624308, −2.736051168203130, −2.409966365545521, −1.687330624506601, −0.9723660104408040, 0, 0.9723660104408040, 1.687330624506601, 2.409966365545521, 2.736051168203130, 3.647405496624308, 4.481563666123548, 4.741381990358685, 5.187527986387865, 6.050894102451226, 6.512456219741193, 6.949381606670196, 7.523248667242851, 8.304175967539516, 8.589617607733836, 9.018108431516341, 9.763377654587806, 10.14048903183742, 10.77520789255849, 11.11699106854549, 11.67420235228204, 12.09001444702613, 12.84005555499916, 13.23575282022768, 13.47773098737986, 14.27341448483639

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