Properties

Degree 4
Conductor $ 2^{5} \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-s − 8-s − 2·9-s − 8·13-s + 16-s + 12·17-s + 2·18-s − 10·25-s + 8·26-s − 12·29-s − 32-s − 12·34-s − 2·36-s + 4·37-s + 12·41-s + 49-s + 10·50-s − 8·52-s + 12·53-s + 12·58-s + 16·61-s + 64-s + 12·68-s + 2·72-s + 4·73-s − 4·74-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 2/3·9-s − 2.21·13-s + 1/4·16-s + 2.91·17-s + 0.471·18-s − 2·25-s + 1.56·26-s − 2.22·29-s − 0.176·32-s − 2.05·34-s − 1/3·36-s + 0.657·37-s + 1.87·41-s + 1/7·49-s + 1.41·50-s − 1.10·52-s + 1.64·53-s + 1.57·58-s + 2.04·61-s + 1/8·64-s + 1.45·68-s + 0.235·72-s + 0.468·73-s − 0.464·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1568\)    =    \(2^{5} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1568} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1568,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.4377085675$
$L(\frac12)$  $\approx$  $0.4377085675$
$L(\frac{3}{2})$   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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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

−13.78118158860551012757665837778, −12.74339932248381599513991172244, −12.30526099005271094386573698884, −11.71064761043096959517781978032, −11.23136141438460806904855801814, −10.11465006559287644470603673937, −9.765547119459919407856461234632, −9.392494157732895222790142944171, −8.182643116839158569055135937831, −7.57571100088867902110310233811, −7.29668061464756137953129595214, −5.62168100149868099368912779374, −5.57928681742950427486583645839, −3.82418745638366191622617161116, −2.48514181668809031419660323366, 2.48514181668809031419660323366, 3.82418745638366191622617161116, 5.57928681742950427486583645839, 5.62168100149868099368912779374, 7.29668061464756137953129595214, 7.57571100088867902110310233811, 8.182643116839158569055135937831, 9.392494157732895222790142944171, 9.765547119459919407856461234632, 10.11465006559287644470603673937, 11.23136141438460806904855801814, 11.71064761043096959517781978032, 12.30526099005271094386573698884, 12.74339932248381599513991172244, 13.78118158860551012757665837778

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