Properties

Label 4-1620e2-1.1-c1e2-0-12
Degree $4$
Conductor $2624400$
Sign $1$
Analytic cond. $167.334$
Root an. cond. $3.59663$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 4·7-s + 3·11-s + 4·13-s + 10·19-s − 6·23-s − 9·29-s − 5·31-s + 4·35-s + 4·37-s − 9·41-s + 10·43-s − 6·47-s + 7·49-s + 24·53-s + 3·55-s + 9·59-s + 10·61-s + 4·65-s − 2·67-s − 6·71-s − 8·73-s + 12·77-s + 4·79-s + 6·83-s + 18·89-s + 16·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s + 0.904·11-s + 1.10·13-s + 2.29·19-s − 1.25·23-s − 1.67·29-s − 0.898·31-s + 0.676·35-s + 0.657·37-s − 1.40·41-s + 1.52·43-s − 0.875·47-s + 49-s + 3.29·53-s + 0.404·55-s + 1.17·59-s + 1.28·61-s + 0.496·65-s − 0.244·67-s − 0.712·71-s − 0.936·73-s + 1.36·77-s + 0.450·79-s + 0.658·83-s + 1.90·89-s + 1.67·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2624400\)    =    \(2^{4} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(167.334\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2624400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.161064876\)
\(L(\frac12)\) \(\approx\) \(4.161064876\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5$C_2$ \( 1 - T + T^{2} \)
good7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.569429984545477459418129512091, −9.208465379889954081069554274531, −8.622256227575664158876007290954, −8.593416531380566037645690054470, −8.037425701110819249561223469302, −7.52724004881275907771391978759, −7.22051565960007170511115964832, −7.04396006342355835317153804657, −6.13151502428225068103718962670, −5.89421356782536761763943563063, −5.53900047128776513251325110132, −5.19057600645540820756634659978, −4.64030612795233785625294822752, −4.06554025235772769589607820847, −3.55726629373921983373685339797, −3.48261453339589512184870212959, −2.29648303199352018562676369560, −2.02783530031818251281040222859, −1.33185388615928719912582820316, −0.907703623212303426226713505128, 0.907703623212303426226713505128, 1.33185388615928719912582820316, 2.02783530031818251281040222859, 2.29648303199352018562676369560, 3.48261453339589512184870212959, 3.55726629373921983373685339797, 4.06554025235772769589607820847, 4.64030612795233785625294822752, 5.19057600645540820756634659978, 5.53900047128776513251325110132, 5.89421356782536761763943563063, 6.13151502428225068103718962670, 7.04396006342355835317153804657, 7.22051565960007170511115964832, 7.52724004881275907771391978759, 8.037425701110819249561223469302, 8.593416531380566037645690054470, 8.622256227575664158876007290954, 9.208465379889954081069554274531, 9.569429984545477459418129512091

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