Properties

Degree 4
Conductor $ 11^{3} \cdot 19^{2} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 8·4-s + 2·5-s − 8·8-s − 5·9-s − 8·10-s + 11-s + 8·13-s − 4·16-s + 20·18-s + 16·20-s − 4·22-s − 2·23-s − 7·25-s − 32·26-s + 32·32-s − 40·36-s − 16·40-s − 16·41-s + 8·44-s − 10·45-s + 8·46-s + 16·47-s − 10·49-s + 28·50-s + 64·52-s + 2·55-s + ⋯
L(s)  = 1  − 2.82·2-s + 4·4-s + 0.894·5-s − 2.82·8-s − 5/3·9-s − 2.52·10-s + 0.301·11-s + 2.21·13-s − 16-s + 4.71·18-s + 3.57·20-s − 0.852·22-s − 0.417·23-s − 7/5·25-s − 6.27·26-s + 5.65·32-s − 6.66·36-s − 2.52·40-s − 2.49·41-s + 1.20·44-s − 1.49·45-s + 1.17·46-s + 2.33·47-s − 1.42·49-s + 3.95·50-s + 8.87·52-s + 0.269·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 480491 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 480491 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(480491\)    =    \(11^{3} \cdot 19^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{480491} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 480491,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{11,\;19\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{11,\;19\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad11$C_1$ \( 1 - T \)
19$C_2$ \( 1 + p T^{2} \)
good2$C_2$ \( ( 1 + p T + p T^{2} )^{2} \)
3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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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

−8.603539619290756001226038948684, −8.271456646397008703308309286986, −7.57431385562905041991919174068, −7.35497151208830631745730081452, −6.38571791919663056094366244020, −6.36261389471308870138602900888, −5.82022083514319120460449276497, −5.31637329804068265476360676633, −4.42771266189087965806263330587, −3.71023016084926681627402828974, −3.06255388161967307495815251846, −2.08501708743688275104095928757, −1.78686819759584429785481099010, −1.00636184286304673122517790617, 0, 1.00636184286304673122517790617, 1.78686819759584429785481099010, 2.08501708743688275104095928757, 3.06255388161967307495815251846, 3.71023016084926681627402828974, 4.42771266189087965806263330587, 5.31637329804068265476360676633, 5.82022083514319120460449276497, 6.36261389471308870138602900888, 6.38571791919663056094366244020, 7.35497151208830631745730081452, 7.57431385562905041991919174068, 8.271456646397008703308309286986, 8.603539619290756001226038948684

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