Properties

Label 2-87120-1.1-c1-0-118
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 − 4·7-s + 2·13-s − 5·17-s + 8·19-s + 5·23-s + 25-s + 6·29-s + 9·31-s − 4·35-s − 4·37-s + 6·41-s + 6·43-s − 13·47-s + 9·49-s − 9·53-s − 10·59-s − 11·61-s + 2·65-s − 12·67-s + 8·71-s + 4·73-s + 5·79-s − 8·83-s − 5·85-s + 8·89-s − 8·91-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-s + 0.554·13-s − 1.21·17-s + 1.83·19-s + 1.04·23-s + 1/5·25-s + 1.11·29-s + 1.61·31-s − 0.676·35-s − 0.657·37-s + 0.937·41-s + 0.914·43-s − 1.89·47-s + 9/7·49-s − 1.23·53-s − 1.30·59-s − 1.40·61-s + 0.248·65-s − 1.46·67-s + 0.949·71-s + 0.468·73-s + 0.562·79-s − 0.878·83-s − 0.542·85-s + 0.847·89-s − 0.838·91-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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−13.97553401457296, −13.61546758802367, −13.22732479288484, −12.77212831357546, −12.21492915764121, −11.77012927353620, −11.09945631516802, −10.64731584525339, −10.13988681177251, −9.481491921405408, −9.323121656930808, −8.835649464099426, −8.065198275764072, −7.556001360051174, −6.817241284840528, −6.382638880554792, −6.232481050040760, −5.371408148565435, −4.811930168760715, −4.270485447250205, −3.354373008200962, −3.000261546900637, −2.615139792615875, −1.500234831788210, −0.9290602946893978, 0, 0.9290602946893978, 1.500234831788210, 2.615139792615875, 3.000261546900637, 3.354373008200962, 4.270485447250205, 4.811930168760715, 5.371408148565435, 6.232481050040760, 6.382638880554792, 6.817241284840528, 7.556001360051174, 8.065198275764072, 8.835649464099426, 9.323121656930808, 9.481491921405408, 10.13988681177251, 10.64731584525339, 11.09945631516802, 11.77012927353620, 12.21492915764121, 12.77212831357546, 13.22732479288484, 13.61546758802367, 13.97553401457296

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