Properties

Degree $4$
Conductor $12446784$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 11-s − 5·13-s − 8·17-s + 5·19-s − 8·23-s + 5·25-s − 3·29-s + 2·31-s + 3·37-s + 6·41-s − 7·43-s − 12·47-s − 11·53-s − 55-s + 5·59-s − 20·61-s + 5·65-s + 7·67-s − 4·71-s + 73-s + 8·79-s + 7·83-s + 8·85-s − 6·89-s − 5·95-s − 25·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.301·11-s − 1.38·13-s − 1.94·17-s + 1.14·19-s − 1.66·23-s + 25-s − 0.557·29-s + 0.359·31-s + 0.493·37-s + 0.937·41-s − 1.06·43-s − 1.75·47-s − 1.51·53-s − 0.134·55-s + 0.650·59-s − 2.56·61-s + 0.620·65-s + 0.855·67-s − 0.474·71-s + 0.117·73-s + 0.900·79-s + 0.768·83-s + 0.867·85-s − 0.635·89-s − 0.512·95-s − 2.53·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12446784\)    =    \(2^{6} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{3528} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 12446784,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - T + 8 T^{2} - p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + 3 T + 46 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 3 T + 62 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 11 T + 122 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 5 T + 110 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 - 7 T + 132 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 - T + 132 T^{2} - p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 7 T + 164 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T + 130 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 25 T + 336 T^{2} + 25 p T^{3} + p^{2} T^{4} \)
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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

−8.199437375520808117057045065771, −8.106745005014989322563050040664, −7.56084715494105884112333032060, −7.45301541763017165200606598950, −6.71654288310414967634349763150, −6.70073632245401744492253588366, −6.23841162056677075120995915228, −5.83356918575981281550840563833, −5.18995316409897141238366334982, −4.85544098925798401277104215023, −4.67468365657476875553358733735, −4.12216009244573645365194056071, −3.78813135785413988787015943187, −3.21602223139018640327563799336, −2.70025665596136581460930474397, −2.38853213020453115209962611679, −1.74294431929387312457142758629, −1.23834149436161297774796709499, 0, 0, 1.23834149436161297774796709499, 1.74294431929387312457142758629, 2.38853213020453115209962611679, 2.70025665596136581460930474397, 3.21602223139018640327563799336, 3.78813135785413988787015943187, 4.12216009244573645365194056071, 4.67468365657476875553358733735, 4.85544098925798401277104215023, 5.18995316409897141238366334982, 5.83356918575981281550840563833, 6.23841162056677075120995915228, 6.70073632245401744492253588366, 6.71654288310414967634349763150, 7.45301541763017165200606598950, 7.56084715494105884112333032060, 8.106745005014989322563050040664, 8.199437375520808117057045065771

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