Properties

Label 2-87120-1.1-c1-0-108
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 − 7-s + 2·13-s + 2·17-s − 7·19-s + 6·23-s + 25-s + 8·29-s + 3·31-s + 35-s + 37-s − 6·41-s − 4·47-s − 6·49-s − 10·53-s + 4·59-s + 11·61-s − 2·65-s + 5·67-s + 6·71-s − 73-s − 79-s + 12·83-s − 2·85-s − 10·89-s − 2·91-s + 7·95-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s + 0.554·13-s + 0.485·17-s − 1.60·19-s + 1.25·23-s + 1/5·25-s + 1.48·29-s + 0.538·31-s + 0.169·35-s + 0.164·37-s − 0.937·41-s − 0.583·47-s − 6/7·49-s − 1.37·53-s + 0.520·59-s + 1.40·61-s − 0.248·65-s + 0.610·67-s + 0.712·71-s − 0.117·73-s − 0.112·79-s + 1.31·83-s − 0.216·85-s − 1.05·89-s − 0.209·91-s + 0.718·95-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 \( 1 + T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−14.25032343282412, −13.58257229844335, −13.03892715789425, −12.75844970894237, −12.23518197743735, −11.65671476337390, −11.15707120154689, −10.73848608813422, −10.16854376284582, −9.755487753166153, −9.059147913545878, −8.516308205485548, −8.218235788357796, −7.680769087318146, −6.742461284611259, −6.621985324586615, −6.148693292967483, −5.140766111828933, −4.927984622720780, −4.118918216805897, −3.653548983997505, −2.984624807546367, −2.484176739154353, −1.544259097987832, −0.8727718110836697, 0, 0.8727718110836697, 1.544259097987832, 2.484176739154353, 2.984624807546367, 3.653548983997505, 4.118918216805897, 4.927984622720780, 5.140766111828933, 6.148693292967483, 6.621985324586615, 6.742461284611259, 7.680769087318146, 8.218235788357796, 8.516308205485548, 9.059147913545878, 9.755487753166153, 10.16854376284582, 10.73848608813422, 11.15707120154689, 11.65671476337390, 12.23518197743735, 12.75844970894237, 13.03892715789425, 13.58257229844335, 14.25032343282412

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