Properties

Degree 4
Conductor $ 2^{12} \cdot 3^{2} \cdot 5^{4} $
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  − 9-s + 4·11-s + 14·19-s − 4·29-s + 10·31-s + 24·41-s + 5·49-s + 12·59-s + 26·61-s + 8·71-s − 16·79-s + 81-s − 32·89-s − 4·99-s + 18·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 17·169-s − 14·171-s + ⋯
L(s)  = 1  − 1/3·9-s + 1.20·11-s + 3.21·19-s − 0.742·29-s + 1.79·31-s + 3.74·41-s + 5/7·49-s + 1.56·59-s + 3.32·61-s + 0.949·71-s − 1.80·79-s + 1/9·81-s − 3.39·89-s − 0.402·99-s + 1.72·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.30·169-s − 1.07·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(23040000\)    =    \(2^{12} \cdot 3^{2} \cdot 5^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4800} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 23040000,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(4.645089894\)
\(L(\frac12)\)  \(\approx\)  \(4.645089894\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 77 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 145 T^{2} + p^{2} T^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.631404241044679861006796995703, −7.938221310560828292654243092291, −7.74695374493879989485486476684, −7.43140856264504493673792403403, −6.93174011971981996490514855232, −6.85219791896510048477784859271, −6.24483117204824828233149814307, −5.84185834318970742589768883747, −5.53250771010055608889464874269, −5.30571021535095644880219082614, −4.84443366446912856076163666733, −4.16798076828796999795368539329, −3.94374071108469997562821317102, −3.72202399456383203643213891536, −2.93671936319750153627805690969, −2.73845633439929779830144635468, −2.34322944440169654584417784809, −1.45423069952354415473069150467, −0.929107553801735562031206855379, −0.809929797341450769054964834201, 0.809929797341450769054964834201, 0.929107553801735562031206855379, 1.45423069952354415473069150467, 2.34322944440169654584417784809, 2.73845633439929779830144635468, 2.93671936319750153627805690969, 3.72202399456383203643213891536, 3.94374071108469997562821317102, 4.16798076828796999795368539329, 4.84443366446912856076163666733, 5.30571021535095644880219082614, 5.53250771010055608889464874269, 5.84185834318970742589768883747, 6.24483117204824828233149814307, 6.85219791896510048477784859271, 6.93174011971981996490514855232, 7.43140856264504493673792403403, 7.74695374493879989485486476684, 7.938221310560828292654243092291, 8.631404241044679861006796995703

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