Properties

Label 2-87120-1.1-c1-0-135
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 + 8·17-s − 2·19-s + 8·23-s + 25-s − 2·35-s + 2·37-s + 6·43-s + 8·47-s − 3·49-s − 6·53-s − 4·59-s − 10·61-s − 2·65-s + 12·67-s + 8·71-s − 10·73-s − 14·79-s − 4·83-s + 8·85-s − 10·89-s + 4·91-s − 2·95-s − 18·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s − 0.554·13-s + 1.94·17-s − 0.458·19-s + 1.66·23-s + 1/5·25-s − 0.338·35-s + 0.328·37-s + 0.914·43-s + 1.16·47-s − 3/7·49-s − 0.824·53-s − 0.520·59-s − 1.28·61-s − 0.248·65-s + 1.46·67-s + 0.949·71-s − 1.17·73-s − 1.57·79-s − 0.439·83-s + 0.867·85-s − 1.05·89-s + 0.419·91-s − 0.205·95-s − 1.82·97-s + 0.0995·101-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 \( 1 - 8 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−14.16247252223831, −13.72640922498215, −12.99637420798759, −12.65954542620637, −12.39092278004134, −11.76152083193952, −11.04736369294641, −10.70273124410825, −10.03827961182234, −9.676927278542398, −9.300991930743279, −8.702636127229907, −8.080680819628889, −7.432982490897488, −7.127014497761163, −6.429240652304143, −5.920395104299951, −5.417922559100925, −4.909571369325853, −4.197223315502601, −3.505331662969790, −2.879253819794876, −2.590933105334899, −1.490601951075889, −0.9935072350026762, 0, 0.9935072350026762, 1.490601951075889, 2.590933105334899, 2.879253819794876, 3.505331662969790, 4.197223315502601, 4.909571369325853, 5.417922559100925, 5.920395104299951, 6.429240652304143, 7.127014497761163, 7.432982490897488, 8.080680819628889, 8.702636127229907, 9.300991930743279, 9.676927278542398, 10.03827961182234, 10.70273124410825, 11.04736369294641, 11.76152083193952, 12.39092278004134, 12.65954542620637, 12.99637420798759, 13.72640922498215, 14.16247252223831

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