Properties

Label 2-87120-1.1-c1-0-106
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 + 2·17-s − 4·19-s − 2·23-s + 25-s − 2·29-s − 8·31-s − 2·35-s + 6·37-s + 10·41-s − 8·43-s − 6·47-s − 3·49-s − 2·53-s + 12·59-s − 2·61-s + 2·65-s + 4·67-s + 6·71-s + 10·79-s − 4·83-s + 2·85-s + 8·89-s − 4·91-s − 4·95-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s + 0.554·13-s + 0.485·17-s − 0.917·19-s − 0.417·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s − 0.338·35-s + 0.986·37-s + 1.56·41-s − 1.21·43-s − 0.875·47-s − 3/7·49-s − 0.274·53-s + 1.56·59-s − 0.256·61-s + 0.248·65-s + 0.488·67-s + 0.712·71-s + 1.12·79-s − 0.439·83-s + 0.216·85-s + 0.847·89-s − 0.419·91-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
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

−14.27852940892926, −13.50520371447765, −13.10173820473664, −12.77224185127140, −12.38748896985988, −11.55230428494557, −11.19169141482737, −10.69512629051556, −10.03281690565772, −9.749745941177040, −9.165966262168959, −8.741370999385792, −8.052593953958594, −7.641122285729947, −6.917182664093456, −6.337064441697074, −6.109043341965174, −5.404389370452147, −4.909542818476783, −4.013447868549162, −3.690253797609631, −3.011183273512010, −2.282873535622824, −1.737727700463738, −0.8648801991010523, 0, 0.8648801991010523, 1.737727700463738, 2.282873535622824, 3.011183273512010, 3.690253797609631, 4.013447868549162, 4.909542818476783, 5.404389370452147, 6.109043341965174, 6.337064441697074, 6.917182664093456, 7.641122285729947, 8.052593953958594, 8.741370999385792, 9.165966262168959, 9.749745941177040, 10.03281690565772, 10.69512629051556, 11.19169141482737, 11.55230428494557, 12.38748896985988, 12.77224185127140, 13.10173820473664, 13.50520371447765, 14.27852940892926

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