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Properties

Label 2-87120-1.1-c1-0-140
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 − 6·13-s + 3·17-s + 4·19-s + 23-s + 25-s − 8·29-s − 5·31-s + 4·35-s − 4·37-s − 2·41-s − 5·47-s + 9·49-s + 13·53-s + 8·59-s + 11·61-s − 6·65-s − 10·67-s − 6·71-s − 4·73-s + 5·79-s − 4·83-s + 3·85-s + 12·89-s − 24·91-s + 4·95-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s − 1.66·13-s + 0.727·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s − 1.48·29-s − 0.898·31-s + 0.676·35-s − 0.657·37-s − 0.312·41-s − 0.729·47-s + 9/7·49-s + 1.78·53-s + 1.04·59-s + 1.40·61-s − 0.744·65-s − 1.22·67-s − 0.712·71-s − 0.468·73-s + 0.562·79-s − 0.439·83-s + 0.325·85-s + 1.27·89-s − 2.51·91-s + 0.410·95-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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 5 T + p T^{2} \)
53 \( 1 - 13 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.40025865728282, −13.64030234552438, −13.26140321434437, −12.63442550507947, −12.04846597219650, −11.70134283417260, −11.30540459744392, −10.65751203961772, −10.05799660617744, −9.837764346922204, −9.004925473433113, −8.805743891109569, −7.871995055250669, −7.613947785712829, −7.213618998248990, −6.601495170015496, −5.568716190863855, −5.276605310942985, −5.119280970840622, −4.244614202709475, −3.685916866170656, −2.862357091699922, −2.216582372465922, −1.710007575616785, −1.051839203126587, 0, 1.051839203126587, 1.710007575616785, 2.216582372465922, 2.862357091699922, 3.685916866170656, 4.244614202709475, 5.119280970840622, 5.276605310942985, 5.568716190863855, 6.601495170015496, 7.213618998248990, 7.613947785712829, 7.871995055250669, 8.805743891109569, 9.004925473433113, 9.837764346922204, 10.05799660617744, 10.65751203961772, 11.30540459744392, 11.70134283417260, 12.04846597219650, 12.63442550507947, 13.26140321434437, 13.64030234552438, 14.40025865728282

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