Properties

Degree 4
Conductor $ 2^{7} \cdot 3^{2} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s − 9-s + 8·11-s + 16-s + 18-s − 8·22-s + 6·25-s − 32-s − 36-s − 8·43-s + 8·44-s − 7·49-s − 6·50-s + 64-s + 8·67-s + 72-s + 81-s + 8·86-s − 8·88-s + 7·98-s − 8·99-s + 6·100-s + 8·107-s + 28·113-s + 26·121-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1/3·9-s + 2.41·11-s + 1/4·16-s + 0.235·18-s − 1.70·22-s + 6/5·25-s − 0.176·32-s − 1/6·36-s − 1.21·43-s + 1.20·44-s − 49-s − 0.848·50-s + 1/8·64-s + 0.977·67-s + 0.117·72-s + 1/9·81-s + 0.862·86-s − 0.852·88-s + 0.707·98-s − 0.804·99-s + 3/5·100-s + 0.773·107-s + 2.63·113-s + 2.36·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56448 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56448 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(56448\)    =    \(2^{7} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{56448} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 56448,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.136111527\)
\(L(\frac12)\)  \(\approx\)  \(1.136111527\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( 1 + T \)
3$C_2$ \( 1 + T^{2} \)
7$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−9.813536793719370156136934900670, −9.488830107252339252415216505294, −8.981242332530629627788661666863, −8.557041619171066391985106539338, −8.196264624071478486989468561724, −7.30425844399090075443129656281, −6.90505066176035016073039144973, −6.41053071455681571589224496085, −6.03012130094571440200692388332, −5.11719413976785294520251081011, −4.46475070828206177816117117812, −3.66600491223085146113555275917, −3.13759619746903620995726706166, −1.97494286469388269114343733065, −1.09691121476639347093183060072, 1.09691121476639347093183060072, 1.97494286469388269114343733065, 3.13759619746903620995726706166, 3.66600491223085146113555275917, 4.46475070828206177816117117812, 5.11719413976785294520251081011, 6.03012130094571440200692388332, 6.41053071455681571589224496085, 6.90505066176035016073039144973, 7.30425844399090075443129656281, 8.196264624071478486989468561724, 8.557041619171066391985106539338, 8.981242332530629627788661666863, 9.488830107252339252415216505294, 9.813536793719370156136934900670

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