Properties

Degree $4$
Conductor $2286144$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 4·7-s + 6·13-s − 4·17-s − 4·19-s − 2·23-s + 5·25-s − 8·29-s − 31-s + 8·35-s + 3·37-s − 16·41-s − 2·43-s − 8·47-s + 9·49-s − 14·53-s − 12·59-s − 61-s − 12·65-s − 3·67-s − 12·71-s − 10·73-s − 79-s − 12·83-s + 8·85-s + 2·89-s − 24·91-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.51·7-s + 1.66·13-s − 0.970·17-s − 0.917·19-s − 0.417·23-s + 25-s − 1.48·29-s − 0.179·31-s + 1.35·35-s + 0.493·37-s − 2.49·41-s − 0.304·43-s − 1.16·47-s + 9/7·49-s − 1.92·53-s − 1.56·59-s − 0.128·61-s − 1.48·65-s − 0.366·67-s − 1.42·71-s − 1.17·73-s − 0.112·79-s − 1.31·83-s + 0.867·85-s + 0.211·89-s − 2.51·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2286144\)    =    \(2^{6} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1512} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2286144,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.08181316739\)
\(L(\frac12)\) \(\approx\) \(0.08181316739\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2^2$ \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
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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

−9.815920094976088700239137358167, −9.172973259470855292700232571058, −8.798819876060741563852154733413, −8.495147465316826516108391790555, −8.305514063770918245731476702495, −7.46415474756975541248291397252, −7.41849947930499806461184205500, −6.59390270747370262162111245295, −6.58158678229828667016807997642, −6.03212921867064587229323274154, −5.86218632339592811783623248602, −4.96540671667323263208021363650, −4.60124758653951207869590886746, −4.04341542234415420150493887393, −3.68115449615579019550638016649, −3.14077846542599206315126721488, −3.02093833265272668689587033858, −1.91087569723324215504985053479, −1.47100699858320982647976328624, −0.11188057592845452424374525565, 0.11188057592845452424374525565, 1.47100699858320982647976328624, 1.91087569723324215504985053479, 3.02093833265272668689587033858, 3.14077846542599206315126721488, 3.68115449615579019550638016649, 4.04341542234415420150493887393, 4.60124758653951207869590886746, 4.96540671667323263208021363650, 5.86218632339592811783623248602, 6.03212921867064587229323274154, 6.58158678229828667016807997642, 6.59390270747370262162111245295, 7.41849947930499806461184205500, 7.46415474756975541248291397252, 8.305514063770918245731476702495, 8.495147465316826516108391790555, 8.798819876060741563852154733413, 9.172973259470855292700232571058, 9.815920094976088700239137358167

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