Properties

Label 2-87120-1.1-c1-0-139
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 + 7-s + 2·13-s − 2·19-s + 25-s + 6·29-s + 4·31-s + 35-s − 4·37-s − 9·41-s + 43-s − 3·47-s − 6·49-s + 6·53-s − 61-s + 2·65-s + 13·67-s − 12·71-s − 16·73-s + 10·79-s + 12·83-s + 3·89-s + 2·91-s − 2·95-s − 10·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 0.554·13-s − 0.458·19-s + 1/5·25-s + 1.11·29-s + 0.718·31-s + 0.169·35-s − 0.657·37-s − 1.40·41-s + 0.152·43-s − 0.437·47-s − 6/7·49-s + 0.824·53-s − 0.128·61-s + 0.248·65-s + 1.58·67-s − 1.42·71-s − 1.87·73-s + 1.12·79-s + 1.31·83-s + 0.317·89-s + 0.209·91-s − 0.205·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-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 \( 1 - T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−14.02287969623273, −13.68125683464184, −13.26232871018859, −12.71560103680186, −12.11144588578211, −11.71878173414437, −11.20647490269208, −10.56322116896899, −10.22359688961008, −9.796652350686883, −8.988481504105807, −8.668751075783251, −8.180843871610974, −7.645613243665281, −6.898299798879820, −6.439312406188483, −6.078519521717430, −5.191014944249878, −4.969849730697410, −4.224103895369679, −3.618626130039505, −2.947353539367366, −2.317545980888073, −1.598499486349677, −1.041080842390955, 0, 1.041080842390955, 1.598499486349677, 2.317545980888073, 2.947353539367366, 3.618626130039505, 4.224103895369679, 4.969849730697410, 5.191014944249878, 6.078519521717430, 6.439312406188483, 6.898299798879820, 7.645613243665281, 8.180843871610974, 8.668751075783251, 8.988481504105807, 9.796652350686883, 10.22359688961008, 10.56322116896899, 11.20647490269208, 11.71878173414437, 12.11144588578211, 12.71560103680186, 13.26232871018859, 13.68125683464184, 14.02287969623273

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