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Properties

Label 2-87120-1.1-c1-0-112
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 + 6·13-s − 3·17-s − 3·23-s + 25-s − 6·29-s + 7·31-s − 8·37-s + 2·41-s + 10·43-s + 3·47-s − 7·49-s − 3·53-s − 2·59-s − 7·61-s − 6·65-s + 12·67-s − 12·71-s − 4·73-s + 3·79-s + 3·85-s + 12·89-s + 16·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.66·13-s − 0.727·17-s − 0.625·23-s + 1/5·25-s − 1.11·29-s + 1.25·31-s − 1.31·37-s + 0.312·41-s + 1.52·43-s + 0.437·47-s − 49-s − 0.412·53-s − 0.260·59-s − 0.896·61-s − 0.744·65-s + 1.46·67-s − 1.42·71-s − 0.468·73-s + 0.337·79-s + 0.325·85-s + 1.27·89-s + 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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: Trivial
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 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−14.07484544892306, −13.66154844469788, −13.15170680922865, −12.78394321828870, −12.04143487410391, −11.73980769476460, −11.10332064979209, −10.74954036430263, −10.36720112727069, −9.550639270750623, −9.038466829351473, −8.709824310174378, −7.970035770869302, −7.798804942755911, −6.926335791862137, −6.508391900804068, −5.941306167425284, −5.501793814609449, −4.619753856289283, −4.219834943123729, −3.612843307858361, −3.143231784106072, −2.286593713581809, −1.621049517366549, −0.8917452431451186, 0, 0.8917452431451186, 1.621049517366549, 2.286593713581809, 3.143231784106072, 3.612843307858361, 4.219834943123729, 4.619753856289283, 5.501793814609449, 5.941306167425284, 6.508391900804068, 6.926335791862137, 7.798804942755911, 7.970035770869302, 8.709824310174378, 9.038466829351473, 9.550639270750623, 10.36720112727069, 10.74954036430263, 11.10332064979209, 11.73980769476460, 12.04143487410391, 12.78394321828870, 13.15170680922865, 13.66154844469788, 14.07484544892306

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