Properties

Degree 4
Conductor $ 2^{2} \cdot 7 \cdot 17 $
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  − 4·3-s + 4-s − 3·7-s + 6·9-s + 6·11-s − 4·12-s − 2·13-s + 16-s + 5·17-s − 2·19-s + 12·21-s − 10·25-s + 4·27-s − 3·28-s − 6·29-s − 8·31-s − 24·33-s + 6·36-s − 2·37-s + 8·39-s + 12·41-s + 16·43-s + 6·44-s − 12·47-s − 4·48-s + 6·49-s − 20·51-s + ⋯
L(s)  = 1  − 2.30·3-s + 1/2·4-s − 1.13·7-s + 2·9-s + 1.80·11-s − 1.15·12-s − 0.554·13-s + 1/4·16-s + 1.21·17-s − 0.458·19-s + 2.61·21-s − 2·25-s + 0.769·27-s − 0.566·28-s − 1.11·29-s − 1.43·31-s − 4.17·33-s + 36-s − 0.328·37-s + 1.28·39-s + 1.87·41-s + 2.43·43-s + 0.904·44-s − 1.75·47-s − 0.577·48-s + 6/7·49-s − 2.80·51-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{476} (1, \cdot )$
Sato-Tate  :  $G_{3,3}$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 476,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.2474290657$
$L(\frac12)$  $\approx$  $0.2474290657$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;17\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;17\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
17$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 6 T + p T^{2} ) \)
good3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−19.929230141, −19.5800270198, −19.1494492613, −18.2120541317, −17.4469801976, −17.2128532162, −16.6317948443, −16.3370810014, −15.7312805996, −14.6077730657, −14.2197849424, −12.8595646329, −12.3052609901, −11.9583441737, −11.2313614144, −10.8489466872, −9.76554711946, −9.28607198826, −7.57571100089, −6.65330731261, −5.99911855172, −5.57928681743, −3.90229547123, 3.90229547123, 5.57928681743, 5.99911855172, 6.65330731261, 7.57571100089, 9.28607198826, 9.76554711946, 10.8489466872, 11.2313614144, 11.9583441737, 12.3052609901, 12.8595646329, 14.2197849424, 14.6077730657, 15.7312805996, 16.3370810014, 16.6317948443, 17.2128532162, 17.4469801976, 18.2120541317, 19.1494492613, 19.5800270198, 19.929230141

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