Properties

Degree 4
Conductor $ 2^{7} \cdot 7^{2} $
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  − 2-s − 4·3-s + 4-s + 4·6-s − 8-s + 6·9-s − 4·12-s + 16-s + 12·17-s − 6·18-s + 4·19-s + 4·24-s − 10·25-s + 4·27-s − 32-s − 12·34-s + 6·36-s − 4·38-s + 12·41-s + 16·43-s − 4·48-s + 49-s + 10·50-s − 48·51-s − 4·54-s − 16·57-s − 12·59-s + ⋯
L(s)  = 1  − 0.707·2-s − 2.30·3-s + 1/2·4-s + 1.63·6-s − 0.353·8-s + 2·9-s − 1.15·12-s + 1/4·16-s + 2.91·17-s − 1.41·18-s + 0.917·19-s + 0.816·24-s − 2·25-s + 0.769·27-s − 0.176·32-s − 2.05·34-s + 36-s − 0.648·38-s + 1.87·41-s + 2.43·43-s − 0.577·48-s + 1/7·49-s + 1.41·50-s − 6.72·51-s − 0.544·54-s − 2.11·57-s − 1.56·59-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(6272\)    =    \(2^{7} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6272} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 6272,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.3095066963$
$L(\frac12)$  $\approx$  $0.3095066963$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( 1 + T \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−12.03047354988920041272999065688, −11.23136141438460806904855801814, −11.08163105431410064794151744331, −10.39623059414136659793705857031, −9.765547119459919407856461234632, −9.479912708108790860766004869191, −8.328602349022309226048251750698, −7.57571100088867902110310233811, −7.29200576171944643329126204500, −6.04992362572003306544038558924, −5.73625972355331222443538857924, −5.57928681742950427486583645839, −4.39126934619586232082133319537, −3.09628357761679869111062402047, −1.00403210582709168291350506092, 1.00403210582709168291350506092, 3.09628357761679869111062402047, 4.39126934619586232082133319537, 5.57928681742950427486583645839, 5.73625972355331222443538857924, 6.04992362572003306544038558924, 7.29200576171944643329126204500, 7.57571100088867902110310233811, 8.328602349022309226048251750698, 9.479912708108790860766004869191, 9.765547119459919407856461234632, 10.39623059414136659793705857031, 11.08163105431410064794151744331, 11.23136141438460806904855801814, 12.03047354988920041272999065688

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