Properties

Degree 4
Conductor $ 2^{5} \cdot 7^{3} $
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  − 2-s − 4·3-s + 4-s + 4·6-s + 7-s − 8-s + 6·9-s − 4·12-s − 14-s + 16-s − 6·18-s + 4·19-s − 4·21-s + 4·24-s − 10·25-s + 4·27-s + 28-s − 12·29-s − 8·31-s − 32-s + 6·36-s + 4·37-s − 4·38-s + 4·42-s − 24·47-s − 4·48-s + 49-s + ⋯
L(s)  = 1  − 0.707·2-s − 2.30·3-s + 1/2·4-s + 1.63·6-s + 0.377·7-s − 0.353·8-s + 2·9-s − 1.15·12-s − 0.267·14-s + 1/4·16-s − 1.41·18-s + 0.917·19-s − 0.872·21-s + 0.816·24-s − 2·25-s + 0.769·27-s + 0.188·28-s − 2.22·29-s − 1.43·31-s − 0.176·32-s + 36-s + 0.657·37-s − 0.648·38-s + 0.617·42-s − 3.50·47-s − 0.577·48-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(10976\)    =    \(2^{5} \cdot 7^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{10976} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 10976,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−11.23136141438460806904855801814, −11.03116828040960387372749944303, −10.02053664953109744946236323249, −9.765547119459919407856461234632, −9.044637307272642732451675861332, −8.199638847074308569053560326614, −7.57571100088867902110310233811, −7.01851678171449461328017406897, −6.24500065333065006116855945568, −5.57928681742950427486583645839, −5.53331414913354333592453982899, −4.55372588498345584946220609716, −3.42554077807776045337947490220, −1.73289286183865826039844070287, 0, 1.73289286183865826039844070287, 3.42554077807776045337947490220, 4.55372588498345584946220609716, 5.53331414913354333592453982899, 5.57928681742950427486583645839, 6.24500065333065006116855945568, 7.01851678171449461328017406897, 7.57571100088867902110310233811, 8.199638847074308569053560326614, 9.044637307272642732451675861332, 9.765547119459919407856461234632, 10.02053664953109744946236323249, 11.03116828040960387372749944303, 11.23136141438460806904855801814

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