Properties

Label 4-3528e2-1.1-c1e2-0-21
Degree $4$
Conductor $12446784$
Sign $1$
Analytic cond. $793.617$
Root an. cond. $5.30765$
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  + 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: $1$
Analytic conductor: \(793.617\)
Root analytic conductor: \(5.30765\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12446784,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.332189330\)
\(L(\frac12)\) \(\approx\) \(3.332189330\)
\(L(\frac{3}{2})\) 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

−8.792818701772083315900052692833, −8.386331879506744594668725255358, −7.907935139283197686345424188363, −7.86759557382819329729202075117, −7.32971537217748960796750790196, −6.77147281483272362346906763760, −6.38773721068119030413837751051, −6.27103055964608145630893214925, −5.65168243120732095731892533736, −5.57956678369545810875355754611, −5.01288076855472667895523495390, −4.56428999046448773875057831504, −3.87303161454762479605286859472, −3.81891220525338338586895157424, −3.31714133186533196958616598180, −2.86509354052728289519145123063, −2.00740241212373459161417113774, −1.89736370255580633887450986083, −1.16876213775196226965374959233, −0.59489909251531065453496301514, 0.59489909251531065453496301514, 1.16876213775196226965374959233, 1.89736370255580633887450986083, 2.00740241212373459161417113774, 2.86509354052728289519145123063, 3.31714133186533196958616598180, 3.81891220525338338586895157424, 3.87303161454762479605286859472, 4.56428999046448773875057831504, 5.01288076855472667895523495390, 5.57956678369545810875355754611, 5.65168243120732095731892533736, 6.27103055964608145630893214925, 6.38773721068119030413837751051, 6.77147281483272362346906763760, 7.32971537217748960796750790196, 7.86759557382819329729202075117, 7.907935139283197686345424188363, 8.386331879506744594668725255358, 8.792818701772083315900052692833

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