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 − 2·5-s − 4·7-s + 9-s + 12-s − 2·15-s + 16-s + 12·17-s − 2·20-s − 4·21-s + 3·25-s + 27-s − 4·28-s + 8·35-s + 36-s + 4·37-s − 12·41-s − 8·43-s − 2·45-s + 48-s + 9·49-s + 12·51-s − 2·60-s − 4·63-s + 64-s − 8·67-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s − 0.894·5-s − 1.51·7-s + 1/3·9-s + 0.288·12-s − 0.516·15-s + 1/4·16-s + 2.91·17-s − 0.447·20-s − 0.872·21-s + 3/5·25-s + 0.192·27-s − 0.755·28-s + 1.35·35-s + 1/6·36-s + 0.657·37-s − 1.87·41-s − 1.21·43-s − 0.298·45-s + 0.144·48-s + 9/7·49-s + 1.68·51-s − 0.258·60-s − 0.503·63-s + 1/8·64-s − 0.977·67-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  =  $(4,\ 132300,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.634533048$
$L(\frac12)$  $\approx$  $1.634533048$
$L(\frac{3}{2})$   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$C_1$ \( ( 1 + T )^{2} \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−9.305587122869086271308905422277, −9.044892677337591862441016284913, −8.145462184294923465930650647398, −7.941231322257043910223245152113, −7.51828953604385192982905489512, −6.97913667513999229030385049043, −6.42217617652666865799421983967, −6.03795573121967783311486434379, −5.26570153655431196939565550355, −4.75701556135232590446975582633, −3.63359445531324431774910445192, −3.36585804145949552210873743484, −3.21826656477469471981988966496, −2.10133345305515366205304847130, −0.879817280069523657260066044025, 0.879817280069523657260066044025, 2.10133345305515366205304847130, 3.21826656477469471981988966496, 3.36585804145949552210873743484, 3.63359445531324431774910445192, 4.75701556135232590446975582633, 5.26570153655431196939565550355, 6.03795573121967783311486434379, 6.42217617652666865799421983967, 6.97913667513999229030385049043, 7.51828953604385192982905489512, 7.941231322257043910223245152113, 8.145462184294923465930650647398, 9.044892677337591862441016284913, 9.305587122869086271308905422277

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