Properties

Label 4-528e2-1.1-c1e2-0-3
Degree $4$
Conductor $278784$
Sign $1$
Analytic cond. $17.7755$
Root an. cond. $2.05331$
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-s − 6·5-s − 2·9-s − 6·15-s + 16·19-s − 6·23-s + 17·25-s − 5·27-s − 20·43-s + 12·45-s − 10·49-s − 12·53-s + 16·57-s − 2·67-s − 6·69-s + 30·71-s − 8·73-s + 17·75-s + 81-s − 96·95-s − 14·97-s + 36·101-s + 36·115-s + 121-s − 18·125-s + 127-s − 20·129-s + ⋯
L(s)  = 1  + 0.577·3-s − 2.68·5-s − 2/3·9-s − 1.54·15-s + 3.67·19-s − 1.25·23-s + 17/5·25-s − 0.962·27-s − 3.04·43-s + 1.78·45-s − 1.42·49-s − 1.64·53-s + 2.11·57-s − 0.244·67-s − 0.722·69-s + 3.56·71-s − 0.936·73-s + 1.96·75-s + 1/9·81-s − 9.84·95-s − 1.42·97-s + 3.58·101-s + 3.35·115-s + 1/11·121-s − 1.60·125-s + 0.0887·127-s − 1.76·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(278784\)    =    \(2^{8} \cdot 3^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(17.7755\)
Root analytic conductor: \(2.05331\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{278784} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 278784,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7929679844\)
\(L(\frac12)\) \(\approx\) \(0.7929679844\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
67$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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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

−8.617818244845431309214438230553, −8.132665807481574488057539990847, −8.053710672355489314643034592341, −7.60852684015633966980596452192, −7.31093235292016540755307260877, −6.70956520147707782074153831968, −6.04886598194608778773456776693, −5.11227132195889762323685903220, −5.09532391743584753214799670039, −4.23465942787126221958004453594, −3.56855668751439570581317008792, −3.27381985666804460223319703873, −3.12033198305302905755997292770, −1.74926942171441723938567252118, −0.51335491910419828067793004742, 0.51335491910419828067793004742, 1.74926942171441723938567252118, 3.12033198305302905755997292770, 3.27381985666804460223319703873, 3.56855668751439570581317008792, 4.23465942787126221958004453594, 5.09532391743584753214799670039, 5.11227132195889762323685903220, 6.04886598194608778773456776693, 6.70956520147707782074153831968, 7.31093235292016540755307260877, 7.60852684015633966980596452192, 8.053710672355489314643034592341, 8.132665807481574488057539990847, 8.617818244845431309214438230553

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