Properties

Label 4-24e4-1.1-c1e2-0-12
Degree $4$
Conductor $331776$
Sign $1$
Analytic cond. $21.1543$
Root an. cond. $2.14461$
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  + 3·5-s − 7-s − 3·9-s − 3·11-s − 13-s + 12·17-s − 8·19-s − 3·23-s + 5·25-s + 3·29-s + 5·31-s − 3·35-s − 4·37-s − 3·41-s + 43-s − 9·45-s − 9·47-s + 7·49-s + 12·53-s − 9·55-s + 3·59-s − 13·61-s + 3·63-s − 3·65-s + 7·67-s + 24·71-s − 20·73-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.377·7-s − 9-s − 0.904·11-s − 0.277·13-s + 2.91·17-s − 1.83·19-s − 0.625·23-s + 25-s + 0.557·29-s + 0.898·31-s − 0.507·35-s − 0.657·37-s − 0.468·41-s + 0.152·43-s − 1.34·45-s − 1.31·47-s + 49-s + 1.64·53-s − 1.21·55-s + 0.390·59-s − 1.66·61-s + 0.377·63-s − 0.372·65-s + 0.855·67-s + 2.84·71-s − 2.34·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(331776\)    =    \(2^{12} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(21.1543\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{576} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 331776,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.865463382\)
\(L(\frac12)\) \(\approx\) \(1.865463382\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
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$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 11 T + 24 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
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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

−10.54316616454141044819462304066, −10.35062920676357758302790618139, −10.31096860861319511967388779965, −9.629299710812925128844189685190, −9.410188143575859190711307273622, −8.566916073724740167392442946972, −8.447248011391483894540889885781, −7.899711178837467910326372702383, −7.49006187641192219307333481718, −6.69694017851654113679538556225, −6.34856182563885755102335982448, −5.77046107631856245557002806581, −5.60836974882995500954408250760, −5.07113302808418267133773026323, −4.50293706671950974106840890426, −3.46573541860135892652635766635, −3.20140224840601163360589573843, −2.38424784963553963887194806924, −1.99699374638027380507059778361, −0.77188997283383854367890158644, 0.77188997283383854367890158644, 1.99699374638027380507059778361, 2.38424784963553963887194806924, 3.20140224840601163360589573843, 3.46573541860135892652635766635, 4.50293706671950974106840890426, 5.07113302808418267133773026323, 5.60836974882995500954408250760, 5.77046107631856245557002806581, 6.34856182563885755102335982448, 6.69694017851654113679538556225, 7.49006187641192219307333481718, 7.899711178837467910326372702383, 8.447248011391483894540889885781, 8.566916073724740167392442946972, 9.410188143575859190711307273622, 9.629299710812925128844189685190, 10.31096860861319511967388779965, 10.35062920676357758302790618139, 10.54316616454141044819462304066

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