Properties

Label 2-87120-1.1-c1-0-104
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 − 4·7-s + 6·13-s − 3·17-s − 4·19-s + 23-s + 25-s + 8·29-s − 5·31-s − 4·35-s − 4·37-s + 2·41-s − 5·47-s + 9·49-s + 13·53-s + 8·59-s − 11·61-s + 6·65-s − 10·67-s − 6·71-s + 4·73-s − 5·79-s + 4·83-s − 3·85-s + 12·89-s − 24·91-s − 4·95-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-s + 1.66·13-s − 0.727·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s + 1.48·29-s − 0.898·31-s − 0.676·35-s − 0.657·37-s + 0.312·41-s − 0.729·47-s + 9/7·49-s + 1.78·53-s + 1.04·59-s − 1.40·61-s + 0.744·65-s − 1.22·67-s − 0.712·71-s + 0.468·73-s − 0.562·79-s + 0.439·83-s − 0.325·85-s + 1.27·89-s − 2.51·91-s − 0.410·95-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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 5 T + p T^{2} \)
53 \( 1 - 13 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−13.85015572561442, −13.62717146925325, −13.15763270740698, −12.84713518366850, −12.27908340102248, −11.72491671468836, −11.06279058325573, −10.48817562581323, −10.37047414915016, −9.638934752401998, −9.031421387167881, −8.761896458591245, −8.332986537672825, −7.443926664654119, −6.790555125305938, −6.483396622735110, −6.050512316144662, −5.583630754965769, −4.745516541217053, −4.096724931814500, −3.586081019912735, −3.029151326968190, −2.408455171252912, −1.649318489948131, −0.8508665134355064, 0, 0.8508665134355064, 1.649318489948131, 2.408455171252912, 3.029151326968190, 3.586081019912735, 4.096724931814500, 4.745516541217053, 5.583630754965769, 6.050512316144662, 6.483396622735110, 6.790555125305938, 7.443926664654119, 8.332986537672825, 8.761896458591245, 9.031421387167881, 9.638934752401998, 10.37047414915016, 10.48817562581323, 11.06279058325573, 11.72491671468836, 12.27908340102248, 12.84713518366850, 13.15763270740698, 13.62717146925325, 13.85015572561442

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