Properties

Degree 4
Conductor $ 2^{2} \cdot 11^{2} \cdot 31^{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  + 4-s − 4·5-s − 6·9-s + 16-s − 4·20-s + 16·23-s + 2·25-s − 2·31-s − 6·36-s + 20·37-s + 24·45-s − 16·47-s − 14·49-s − 12·53-s − 24·59-s + 64-s − 24·67-s + 16·71-s − 4·80-s + 27·81-s − 12·89-s + 16·92-s + 4·97-s + 2·100-s + 16·103-s + 4·113-s − 64·115-s + ⋯
L(s)  = 1  + 1/2·4-s − 1.78·5-s − 2·9-s + 1/4·16-s − 0.894·20-s + 3.33·23-s + 2/5·25-s − 0.359·31-s − 36-s + 3.28·37-s + 3.57·45-s − 2.33·47-s − 2·49-s − 1.64·53-s − 3.12·59-s + 1/8·64-s − 2.93·67-s + 1.89·71-s − 0.447·80-s + 3·81-s − 1.27·89-s + 1.66·92-s + 0.406·97-s + 1/5·100-s + 1.57·103-s + 0.376·113-s − 5.96·115-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 465124 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 465124 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(465124\)    =    \(2^{2} \cdot 11^{2} \cdot 31^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{465124} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 465124,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.7485291848$
$L(\frac12)$  $\approx$  $0.7485291848$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;11,\;31\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;11,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_2$ \( 1 + p T^{2} \)
31$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−8.366950675915466890526830696713, −7.972117516729530841460864476054, −7.76857292337562945056886541715, −7.44813954497716788319038433151, −6.62362054253814874759895013405, −6.32858579420662158347624019611, −5.90000984791265077671128060122, −5.12409663272380286045429532906, −4.74203912193695697715499891246, −4.30914563361016729079940273832, −3.22916369280236661872194294152, −3.16188554299090604324575557253, −2.86951068240353434231414371750, −1.61564515331644559705016592088, −0.45818161262278296890065564652, 0.45818161262278296890065564652, 1.61564515331644559705016592088, 2.86951068240353434231414371750, 3.16188554299090604324575557253, 3.22916369280236661872194294152, 4.30914563361016729079940273832, 4.74203912193695697715499891246, 5.12409663272380286045429532906, 5.90000984791265077671128060122, 6.32858579420662158347624019611, 6.62362054253814874759895013405, 7.44813954497716788319038433151, 7.76857292337562945056886541715, 7.972117516729530841460864476054, 8.366950675915466890526830696713

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