Properties

Degree 4
Conductor $ 2^{4} \cdot 5 \cdot 13 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 5-s − 2·9-s − 3·13-s + 4·16-s + 6·17-s + 2·20-s − 4·25-s + 4·36-s + 4·37-s + 6·41-s + 2·45-s − 4·49-s + 6·52-s − 20·61-s − 8·64-s + 3·65-s − 12·68-s + 4·73-s − 4·80-s − 5·81-s − 6·85-s + 12·89-s − 2·97-s + 8·100-s − 14·109-s + 18·113-s + ⋯
L(s)  = 1  − 4-s − 0.447·5-s − 2/3·9-s − 0.832·13-s + 16-s + 1.45·17-s + 0.447·20-s − 4/5·25-s + 2/3·36-s + 0.657·37-s + 0.937·41-s + 0.298·45-s − 4/7·49-s + 0.832·52-s − 2.56·61-s − 64-s + 0.372·65-s − 1.45·68-s + 0.468·73-s − 0.447·80-s − 5/9·81-s − 0.650·85-s + 1.27·89-s − 0.203·97-s + 4/5·100-s − 1.34·109-s + 1.69·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1040} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1040,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.4252183223$
$L(\frac12)$  $\approx$  $0.4252183223$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;13\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + p T^{2} \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−14.13699022915792173969386231889, −13.73132384905208795991781869978, −12.83163059852659156555701285288, −12.29746181263507627624712512543, −11.78071801992459051606911551036, −10.94156059915506027778058563467, −10.06351729877973795451488539436, −9.546006320300178143285882286771, −8.824516499451245410039466604829, −7.87114926676923035162899791276, −7.60387813835405712032643111432, −6.11302103058811434541819221939, −5.32475119137192445904899875328, −4.35420037564796542461231717693, −3.19803865538527220934460800982, 3.19803865538527220934460800982, 4.35420037564796542461231717693, 5.32475119137192445904899875328, 6.11302103058811434541819221939, 7.60387813835405712032643111432, 7.87114926676923035162899791276, 8.824516499451245410039466604829, 9.546006320300178143285882286771, 10.06351729877973795451488539436, 10.94156059915506027778058563467, 11.78071801992459051606911551036, 12.29746181263507627624712512543, 12.83163059852659156555701285288, 13.73132384905208795991781869978, 14.13699022915792173969386231889

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