Properties

Degree $2$
Conductor $87120$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4·7-s + 4·13-s − 4·19-s − 6·23-s + 25-s − 6·29-s − 8·31-s + 4·35-s + 2·37-s + 6·41-s + 8·43-s + 6·47-s + 9·49-s + 6·53-s − 12·59-s − 2·61-s − 4·65-s + 10·67-s − 12·71-s + 16·73-s + 8·79-s − 6·89-s − 16·91-s + 4·95-s + 14·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s + 1.10·13-s − 0.917·19-s − 1.25·23-s + 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.676·35-s + 0.328·37-s + 0.937·41-s + 1.21·43-s + 0.875·47-s + 9/7·49-s + 0.824·53-s − 1.56·59-s − 0.256·61-s − 0.496·65-s + 1.22·67-s − 1.42·71-s + 1.87·73-s + 0.900·79-s − 0.635·89-s − 1.67·91-s + 0.410·95-s + 1.42·97-s + 0.0995·101-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: $1$
Motivic weight: \(1\)
Character: $\chi_{87120} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 87120,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6414894341\)
\(L(\frac12)\) \(\approx\) \(0.6414894341\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−13.83779814703676, −13.34279230106179, −12.81552375093620, −12.55268745752804, −12.09001931243545, −11.33506513580107, −10.80587488488927, −10.63699196615051, −9.814517853778296, −9.370175908890980, −8.974620077787424, −8.450479317740666, −7.685394615646233, −7.410468111033469, −6.659298498155139, −6.086060775561553, −5.947582043121594, −5.188068305376285, −4.172941082692343, −3.855156100296041, −3.541621221620832, −2.649594639108158, −2.131679509416876, −1.181393891306063, −0.2716308894760162, 0.2716308894760162, 1.181393891306063, 2.131679509416876, 2.649594639108158, 3.541621221620832, 3.855156100296041, 4.172941082692343, 5.188068305376285, 5.947582043121594, 6.086060775561553, 6.659298498155139, 7.410468111033469, 7.685394615646233, 8.450479317740666, 8.974620077787424, 9.370175908890980, 9.814517853778296, 10.63699196615051, 10.80587488488927, 11.33506513580107, 12.09001931243545, 12.55268745752804, 12.81552375093620, 13.34279230106179, 13.83779814703676

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