Properties

Label 4-78e2-1.1-c1e2-0-0
Degree $4$
Conductor $6084$
Sign $1$
Analytic cond. $0.387921$
Root an. cond. $0.789197$
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  − 3·3-s + 4-s + 2·7-s + 6·9-s − 3·12-s − 2·13-s + 16-s + 12·19-s − 6·21-s − 9·25-s − 9·27-s + 2·28-s + 8·31-s + 6·36-s + 6·37-s + 6·39-s − 10·43-s − 3·48-s − 11·49-s − 2·52-s − 36·57-s − 16·61-s + 12·63-s + 64-s − 4·67-s − 20·73-s + 27·75-s + ⋯
L(s)  = 1  − 1.73·3-s + 1/2·4-s + 0.755·7-s + 2·9-s − 0.866·12-s − 0.554·13-s + 1/4·16-s + 2.75·19-s − 1.30·21-s − 9/5·25-s − 1.73·27-s + 0.377·28-s + 1.43·31-s + 36-s + 0.986·37-s + 0.960·39-s − 1.52·43-s − 0.433·48-s − 1.57·49-s − 0.277·52-s − 4.76·57-s − 2.04·61-s + 1.51·63-s + 1/8·64-s − 0.488·67-s − 2.34·73-s + 3.11·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6084\)    =    \(2^{2} \cdot 3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.387921\)
Root analytic conductor: \(0.789197\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6084,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6467806837\)
\(L(\frac12)\) \(\approx\) \(0.6467806837\)
\(L(\frac{3}{2})\) 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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_2$ \( 1 + p T + p T^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−11.86689967980337109441141028229, −11.49712673239204411601833482255, −11.27108277800122947493852923485, −10.28025762974353066563394220777, −9.945515510046352754787747239151, −9.474403868218177338841530557687, −8.251376244266523722285240612658, −7.45985560667588965978165906144, −7.34105417162701727305699109547, −6.18070949711225184510895261552, −5.88908450915949561478136129218, −4.98797766425562785919630567490, −4.61153453491182141362175201622, −3.20294866149791344283024452913, −1.48472360333277511919704091526, 1.48472360333277511919704091526, 3.20294866149791344283024452913, 4.61153453491182141362175201622, 4.98797766425562785919630567490, 5.88908450915949561478136129218, 6.18070949711225184510895261552, 7.34105417162701727305699109547, 7.45985560667588965978165906144, 8.251376244266523722285240612658, 9.474403868218177338841530557687, 9.945515510046352754787747239151, 10.28025762974353066563394220777, 11.27108277800122947493852923485, 11.49712673239204411601833482255, 11.86689967980337109441141028229

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