Properties

Label 2-87120-1.1-c1-0-105
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 − 3·7-s + 5·13-s + 5·17-s + 19-s − 4·23-s + 25-s − 5·29-s − 2·31-s + 3·35-s + 3·37-s + 2·41-s + 2·43-s + 6·47-s + 2·49-s + 4·59-s − 8·61-s − 5·65-s − 6·67-s + 3·71-s + 10·73-s − 10·79-s − 9·83-s − 5·85-s + 12·89-s − 15·91-s − 95-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.13·7-s + 1.38·13-s + 1.21·17-s + 0.229·19-s − 0.834·23-s + 1/5·25-s − 0.928·29-s − 0.359·31-s + 0.507·35-s + 0.493·37-s + 0.312·41-s + 0.304·43-s + 0.875·47-s + 2/7·49-s + 0.520·59-s − 1.02·61-s − 0.620·65-s − 0.733·67-s + 0.356·71-s + 1.17·73-s − 1.12·79-s − 0.987·83-s − 0.542·85-s + 1.27·89-s − 1.57·91-s − 0.102·95-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 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
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

−14.05884812496676, −13.66858945529243, −13.06421477992544, −12.74943632480303, −12.17087513443830, −11.77890471934286, −11.10916204739154, −10.75084155921547, −10.12862188051855, −9.636904225791159, −9.230091272814426, −8.608852324917584, −8.107781119282476, −7.510160532590540, −7.143764384162019, −6.343906656362119, −5.932036468238867, −5.621729713027470, −4.749425217427834, −3.935381251812805, −3.663849643842154, −3.159667865587244, −2.447626994581324, −1.520185582967745, −0.8569309386162201, 0, 0.8569309386162201, 1.520185582967745, 2.447626994581324, 3.159667865587244, 3.663849643842154, 3.935381251812805, 4.749425217427834, 5.621729713027470, 5.932036468238867, 6.343906656362119, 7.143764384162019, 7.510160532590540, 8.107781119282476, 8.608852324917584, 9.230091272814426, 9.636904225791159, 10.12862188051855, 10.75084155921547, 11.10916204739154, 11.77890471934286, 12.17087513443830, 12.74943632480303, 13.06421477992544, 13.66858945529243, 14.05884812496676

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