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  + 4·5-s − 6·9-s + 4·13-s − 12·17-s + 2·25-s + 12·29-s − 4·37-s + 4·41-s − 24·45-s + 49-s + 12·53-s − 12·61-s + 16·65-s + 20·73-s + 27·81-s − 48·85-s − 12·89-s − 12·97-s + 4·101-s − 20·109-s + 4·113-s − 24·117-s − 6·121-s − 28·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1.78·5-s − 2·9-s + 1.10·13-s − 2.91·17-s + 2/5·25-s + 2.22·29-s − 0.657·37-s + 0.624·41-s − 3.57·45-s + 1/7·49-s + 1.64·53-s − 1.53·61-s + 1.98·65-s + 2.34·73-s + 3·81-s − 5.20·85-s − 1.27·89-s − 1.21·97-s + 0.398·101-s − 1.91·109-s + 0.376·113-s − 2.21·117-s − 0.545·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-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  =  $(4,\ 6272,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.004073929$
$L(\frac12)$  $\approx$  $1.004073929$
$L(\frac{3}{2})$   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 \( 1 \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−11.84146297682028690633889717654, −11.40509490101706296629658789662, −10.63587746243635306398043641619, −10.60224244605800499935670328540, −9.457349313423188294666539437989, −9.117894034638208548514808203742, −8.560607806232820418165457845354, −8.188420417684287628984093773695, −6.67949194679031572918581545842, −6.42262084813563967264199691637, −5.82613210142107824134972732669, −5.22316409435338364257345412913, −4.18465932715971087267449515138, −2.79183800612725741835263061431, −2.14556791802447766709825303506, 2.14556791802447766709825303506, 2.79183800612725741835263061431, 4.18465932715971087267449515138, 5.22316409435338364257345412913, 5.82613210142107824134972732669, 6.42262084813563967264199691637, 6.67949194679031572918581545842, 8.188420417684287628984093773695, 8.560607806232820418165457845354, 9.117894034638208548514808203742, 9.457349313423188294666539437989, 10.60224244605800499935670328540, 10.63587746243635306398043641619, 11.40509490101706296629658789662, 11.84146297682028690633889717654

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