Properties

Label 4-219e2-1.1-c1e2-0-6
Degree $4$
Conductor $47961$
Sign $-1$
Analytic cond. $3.05803$
Root an. cond. $1.32239$
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  + 2·3-s − 4-s − 6·7-s + 9-s − 2·12-s − 3·16-s − 4·19-s − 12·21-s − 2·25-s − 4·27-s + 6·28-s + 6·31-s − 36-s − 4·37-s − 6·43-s − 6·48-s + 14·49-s − 8·57-s − 8·61-s − 6·63-s + 7·64-s + 4·67-s − 2·73-s − 4·75-s + 4·76-s − 4·79-s − 11·81-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s − 2.26·7-s + 1/3·9-s − 0.577·12-s − 3/4·16-s − 0.917·19-s − 2.61·21-s − 2/5·25-s − 0.769·27-s + 1.13·28-s + 1.07·31-s − 1/6·36-s − 0.657·37-s − 0.914·43-s − 0.866·48-s + 2·49-s − 1.05·57-s − 1.02·61-s − 0.755·63-s + 7/8·64-s + 0.488·67-s − 0.234·73-s − 0.461·75-s + 0.458·76-s − 0.450·79-s − 1.22·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47961 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(47961\)    =    \(3^{2} \cdot 73^{2}\)
Sign: $-1$
Analytic conductor: \(3.05803\)
Root analytic conductor: \(1.32239\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 47961,\ (\ :1/2, 1/2),\ -1)\)

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$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ \( 1 - 2 T + p T^{2} \)
73$C_2$ \( 1 + 2 T + p T^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
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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

−9.663477846467172549479061889828, −9.396152038274767786766575447619, −8.945834811056945865824819305285, −8.422569047301544181563986321416, −8.026898874215042644635425441906, −7.16362665857073601207380088132, −6.70054833293196549592528243866, −6.27685355807958818406186977112, −5.64843217071871487921675422735, −4.67343670052114048280588496244, −4.03655616886073847473994412666, −3.36502920699158770798746585282, −2.94080839143346445109060006209, −2.11550072184673889865559323192, 0, 2.11550072184673889865559323192, 2.94080839143346445109060006209, 3.36502920699158770798746585282, 4.03655616886073847473994412666, 4.67343670052114048280588496244, 5.64843217071871487921675422735, 6.27685355807958818406186977112, 6.70054833293196549592528243866, 7.16362665857073601207380088132, 8.026898874215042644635425441906, 8.422569047301544181563986321416, 8.945834811056945865824819305285, 9.396152038274767786766575447619, 9.663477846467172549479061889828

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