Properties

Label 2-87120-1.1-c1-0-14
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 6·13-s − 7·17-s + 4·19-s + 3·23-s + 25-s − 4·29-s − 3·31-s + 2·41-s − 4·43-s − 7·47-s − 7·49-s + 5·53-s + 4·59-s + 61-s − 6·65-s − 2·67-s − 10·71-s − 4·73-s + 5·79-s + 12·83-s − 7·85-s + 4·95-s + 6·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.66·13-s − 1.69·17-s + 0.917·19-s + 0.625·23-s + 1/5·25-s − 0.742·29-s − 0.538·31-s + 0.312·41-s − 0.609·43-s − 1.02·47-s − 49-s + 0.686·53-s + 0.520·59-s + 0.128·61-s − 0.744·65-s − 0.244·67-s − 1.18·71-s − 0.468·73-s + 0.562·79-s + 1.31·83-s − 0.759·85-s + 0.410·95-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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: \(0\)
Selberg data: \((2,\ 87120,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.036308191\)
\(L(\frac12)\) \(\approx\) \(1.036308191\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−13.72543621200102, −13.45445005568990, −12.95552263321042, −12.50898541069096, −11.89406075704013, −11.39788282357648, −11.03653780851827, −10.31475769059574, −9.923933502718397, −9.293439179987602, −9.123170919282147, −8.410447218150696, −7.726355863502558, −7.242906095752530, −6.833109103869335, −6.279795054286720, −5.564426230802487, −4.960924498577047, −4.752443510267637, −3.933752968017773, −3.216760227281059, −2.559934877321254, −2.091578143303468, −1.391698062584764, −0.3140407624396482, 0.3140407624396482, 1.391698062584764, 2.091578143303468, 2.559934877321254, 3.216760227281059, 3.933752968017773, 4.752443510267637, 4.960924498577047, 5.564426230802487, 6.279795054286720, 6.833109103869335, 7.242906095752530, 7.726355863502558, 8.410447218150696, 9.123170919282147, 9.293439179987602, 9.923933502718397, 10.31475769059574, 11.03653780851827, 11.39788282357648, 11.89406075704013, 12.50898541069096, 12.95552263321042, 13.45445005568990, 13.72543621200102

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