Properties

Label 2-87120-1.1-c1-0-141
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 + 2·7-s − 4·13-s + 2·17-s − 2·19-s + 4·23-s + 25-s + 6·29-s + 2·35-s + 10·37-s + 2·41-s − 2·43-s − 3·49-s − 6·53-s − 4·61-s − 4·65-s − 4·67-s − 2·79-s + 2·85-s − 10·89-s − 8·91-s − 2·95-s − 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.755·7-s − 1.10·13-s + 0.485·17-s − 0.458·19-s + 0.834·23-s + 1/5·25-s + 1.11·29-s + 0.338·35-s + 1.64·37-s + 0.312·41-s − 0.304·43-s − 3/7·49-s − 0.824·53-s − 0.512·61-s − 0.496·65-s − 0.488·67-s − 0.225·79-s + 0.216·85-s − 1.05·89-s − 0.838·91-s − 0.205·95-s − 0.609·97-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: $\chi_{87120} (1, \cdot )$
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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−14.24353850424782, −13.72602419634840, −13.10912980607374, −12.68111751023674, −12.20723646223404, −11.69844299637288, −11.11929488869990, −10.75931779302539, −10.09088175914645, −9.690937671763511, −9.255239627908680, −8.523450453515351, −8.161668863928577, −7.518349404593274, −7.154453565084193, −6.378722982871617, −6.015684974440648, −5.227298665625352, −4.799903057846999, −4.447432967257053, −3.599367073104707, −2.713568704825757, −2.523214122078426, −1.557812583703497, −1.053605269612925, 0, 1.053605269612925, 1.557812583703497, 2.523214122078426, 2.713568704825757, 3.599367073104707, 4.447432967257053, 4.799903057846999, 5.227298665625352, 6.015684974440648, 6.378722982871617, 7.154453565084193, 7.518349404593274, 8.161668863928577, 8.523450453515351, 9.255239627908680, 9.690937671763511, 10.09088175914645, 10.75931779302539, 11.11929488869990, 11.69844299637288, 12.20723646223404, 12.68111751023674, 13.10912980607374, 13.72602419634840, 14.24353850424782

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