Properties

Label 2-87120-1.1-c1-0-115
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 − 2·17-s + 7·19-s + 6·23-s + 25-s − 8·29-s + 3·31-s − 35-s + 37-s + 6·41-s − 4·47-s − 6·49-s − 10·53-s + 4·59-s − 11·61-s + 2·65-s + 5·67-s + 6·71-s + 73-s + 79-s − 12·83-s + 2·85-s − 10·89-s − 2·91-s − 7·95-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 0.554·13-s − 0.485·17-s + 1.60·19-s + 1.25·23-s + 1/5·25-s − 1.48·29-s + 0.538·31-s − 0.169·35-s + 0.164·37-s + 0.937·41-s − 0.583·47-s − 6/7·49-s − 1.37·53-s + 0.520·59-s − 1.40·61-s + 0.248·65-s + 0.610·67-s + 0.712·71-s + 0.117·73-s + 0.112·79-s − 1.31·83-s + 0.216·85-s − 1.05·89-s − 0.209·91-s − 0.718·95-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: 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 \( 1 - T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−14.31979633016342, −13.64993376187751, −13.12181197598779, −12.65154943198193, −12.23630086937228, −11.47416037802681, −11.22541944379962, −10.98010028830194, −10.01617467141910, −9.680457474373905, −9.167755733256785, −8.669317132989238, −7.962094960294143, −7.553100523934415, −7.192071741817243, −6.559281369821055, −5.880859254318238, −5.261752047930851, −4.801303997911153, −4.345781133524036, −3.459255210052786, −3.116970739963857, −2.371759992867843, −1.578850407892284, −0.9088450905320447, 0, 0.9088450905320447, 1.578850407892284, 2.371759992867843, 3.116970739963857, 3.459255210052786, 4.345781133524036, 4.801303997911153, 5.261752047930851, 5.880859254318238, 6.559281369821055, 7.192071741817243, 7.553100523934415, 7.962094960294143, 8.669317132989238, 9.167755733256785, 9.680457474373905, 10.01617467141910, 10.98010028830194, 11.22541944379962, 11.47416037802681, 12.23630086937228, 12.65154943198193, 13.12181197598779, 13.64993376187751, 14.31979633016342

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