Properties

Label 2-87120-1.1-c1-0-137
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 − 2·13-s − 2·17-s + 8·19-s + 4·23-s + 25-s + 2·29-s − 8·31-s − 2·37-s + 6·41-s + 8·43-s − 4·47-s − 7·49-s − 2·53-s + 4·59-s + 6·61-s − 2·65-s + 12·67-s − 12·71-s − 2·73-s − 4·83-s − 2·85-s + 6·89-s + 8·95-s − 14·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.554·13-s − 0.485·17-s + 1.83·19-s + 0.834·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s − 0.328·37-s + 0.937·41-s + 1.21·43-s − 0.583·47-s − 49-s − 0.274·53-s + 0.520·59-s + 0.768·61-s − 0.248·65-s + 1.46·67-s − 1.42·71-s − 0.234·73-s − 0.439·83-s − 0.216·85-s + 0.635·89-s + 0.820·95-s − 1.42·97-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: \(1\)
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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−14.16676951623200, −13.64165847555372, −13.19319690690340, −12.66193203744180, −12.28957625980311, −11.60354044015404, −11.16338535862072, −10.78742672671083, −10.05709407024840, −9.590315848320436, −9.268102820492955, −8.762068731939020, −8.003300462153283, −7.551930451660907, −6.986967554300816, −6.633513737270221, −5.773278371810284, −5.361843128794542, −4.969469778952626, −4.209701712648772, −3.584956040035517, −2.872722504312467, −2.461002402939388, −1.563896237276592, −1.002797341066645, 0, 1.002797341066645, 1.563896237276592, 2.461002402939388, 2.872722504312467, 3.584956040035517, 4.209701712648772, 4.969469778952626, 5.361843128794542, 5.773278371810284, 6.633513737270221, 6.986967554300816, 7.551930451660907, 8.003300462153283, 8.762068731939020, 9.268102820492955, 9.590315848320436, 10.05709407024840, 10.78742672671083, 11.16338535862072, 11.60354044015404, 12.28957625980311, 12.66193203744180, 13.19319690690340, 13.64165847555372, 14.16676951623200

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