Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{3} \cdot 5^{2} \cdot 7^{2} $
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  + 3-s + 4-s + 2·7-s + 9-s + 12-s + 4·13-s + 16-s + 16·19-s + 2·21-s + 25-s + 27-s + 2·28-s − 8·31-s + 36-s − 20·37-s + 4·39-s − 8·43-s + 48-s + 3·49-s + 4·52-s + 16·57-s − 20·61-s + 2·63-s + 64-s − 8·67-s − 20·73-s + 75-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s + 0.288·12-s + 1.10·13-s + 1/4·16-s + 3.67·19-s + 0.436·21-s + 1/5·25-s + 0.192·27-s + 0.377·28-s − 1.43·31-s + 1/6·36-s − 3.28·37-s + 0.640·39-s − 1.21·43-s + 0.144·48-s + 3/7·49-s + 0.554·52-s + 2.11·57-s − 2.56·61-s + 0.251·63-s + 1/8·64-s − 0.977·67-s − 2.34·73-s + 0.115·75-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 132300 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 132300 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(132300\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{132300} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 132300,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $2.747007217$
$L(\frac12)$  $\approx$  $2.747007217$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( 1 - T \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( ( 1 - T )^{2} \)
good11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−9.262397060563074490443203035163, −8.790922026040486061843055845413, −8.581528352643131576230344729464, −7.67845508591579519153392227466, −7.48230039465939347417283743898, −7.19801386358340821477645347390, −6.43968643454495280602009146366, −5.76093596813527562032392850894, −5.20272031164140682979668587117, −4.97103303999261607614028493172, −3.88273199773070623460496754838, −3.22977772739188491587716942834, −3.13125557587754062819456448397, −1.69594451483764738863265937021, −1.41006155845401754086894809680, 1.41006155845401754086894809680, 1.69594451483764738863265937021, 3.13125557587754062819456448397, 3.22977772739188491587716942834, 3.88273199773070623460496754838, 4.97103303999261607614028493172, 5.20272031164140682979668587117, 5.76093596813527562032392850894, 6.43968643454495280602009146366, 7.19801386358340821477645347390, 7.48230039465939347417283743898, 7.67845508591579519153392227466, 8.581528352643131576230344729464, 8.790922026040486061843055845413, 9.262397060563074490443203035163

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