Properties

Degree 4
Conductor $ 2^{6} \cdot 53^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 6·3-s − 4-s + 6·6-s + 3·8-s + 21·9-s + 6·12-s − 16-s − 6·17-s − 21·18-s − 10·19-s − 18·24-s − 10·25-s − 54·27-s − 5·32-s + 6·34-s − 21·36-s + 10·38-s + 12·41-s − 4·43-s + 6·48-s + 2·49-s + 10·50-s + 36·51-s + 54·54-s + 60·57-s − 4·59-s + ⋯
L(s)  = 1  − 0.707·2-s − 3.46·3-s − 1/2·4-s + 2.44·6-s + 1.06·8-s + 7·9-s + 1.73·12-s − 1/4·16-s − 1.45·17-s − 4.94·18-s − 2.29·19-s − 3.67·24-s − 2·25-s − 10.3·27-s − 0.883·32-s + 1.02·34-s − 7/2·36-s + 1.62·38-s + 1.87·41-s − 0.609·43-s + 0.866·48-s + 2/7·49-s + 1.41·50-s + 5.04·51-s + 7.34·54-s + 7.94·57-s − 0.520·59-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 179776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 179776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(179776\)    =    \(2^{6} \cdot 53^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{179776} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 179776,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−8.820142471571876337599747567018, −8.205174103101839964623123933331, −7.40585307678501251492741626335, −7.20736947999270954805514319970, −6.49177047979490854052512324082, −6.04347889941024738587947019148, −5.95808888910984261709863888066, −5.27863549534731961075900677576, −4.50862835068296220400083952922, −4.42654752979487491573226444690, −4.04318522706247983721491253685, −2.12271179066542804694951246257, −1.33598343949561168216183034148, 0, 0, 1.33598343949561168216183034148, 2.12271179066542804694951246257, 4.04318522706247983721491253685, 4.42654752979487491573226444690, 4.50862835068296220400083952922, 5.27863549534731961075900677576, 5.95808888910984261709863888066, 6.04347889941024738587947019148, 6.49177047979490854052512324082, 7.20736947999270954805514319970, 7.40585307678501251492741626335, 8.205174103101839964623123933331, 8.820142471571876337599747567018

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