Properties

Label 2-87120-1.1-c1-0-11
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 − 4·7-s − 4·13-s + 4·19-s − 8·23-s + 25-s − 4·29-s − 4·31-s − 4·35-s − 2·37-s + 4·41-s + 12·43-s − 8·47-s + 9·49-s + 10·53-s − 4·65-s + 4·67-s + 4·71-s + 4·73-s − 12·79-s − 16·83-s + 6·89-s + 16·91-s + 4·95-s − 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-s − 1.10·13-s + 0.917·19-s − 1.66·23-s + 1/5·25-s − 0.742·29-s − 0.718·31-s − 0.676·35-s − 0.328·37-s + 0.624·41-s + 1.82·43-s − 1.16·47-s + 9/7·49-s + 1.37·53-s − 0.496·65-s + 0.488·67-s + 0.474·71-s + 0.468·73-s − 1.35·79-s − 1.75·83-s + 0.635·89-s + 1.67·91-s + 0.410·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-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

\(L(1)\) \(\approx\) \(0.7380575314\)
\(L(\frac12)\) \(\approx\) \(0.7380575314\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
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

−13.99263075946505, −13.31595707495196, −12.85376235149211, −12.56705290553046, −12.00029731017844, −11.54378328729616, −10.86570164557083, −10.10993356055010, −9.999022788239830, −9.440677314836233, −9.140872346207686, −8.421802965523278, −7.617760624830063, −7.331436144903252, −6.784621999393206, −6.093553827655698, −5.759537499712075, −5.250814505153204, −4.451736432670748, −3.785814473630532, −3.368885443427463, −2.514952892390691, −2.257650733308267, −1.251316685925353, −0.2774765693331784, 0.2774765693331784, 1.251316685925353, 2.257650733308267, 2.514952892390691, 3.368885443427463, 3.785814473630532, 4.451736432670748, 5.250814505153204, 5.759537499712075, 6.093553827655698, 6.784621999393206, 7.331436144903252, 7.617760624830063, 8.421802965523278, 9.140872346207686, 9.440677314836233, 9.999022788239830, 10.10993356055010, 10.86570164557083, 11.54378328729616, 12.00029731017844, 12.56705290553046, 12.85376235149211, 13.31595707495196, 13.99263075946505

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