Properties

Degree 4
Conductor $ 2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13^{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  − 2·2-s + 3·4-s + 2·7-s − 4·8-s − 2·9-s − 4·13-s − 4·14-s + 5·16-s + 4·18-s − 5·25-s + 8·26-s + 6·28-s − 12·29-s − 6·32-s − 6·36-s + 4·37-s − 24·47-s + 3·49-s + 10·50-s − 12·52-s − 8·56-s + 24·58-s + 16·61-s − 4·63-s + 7·64-s − 8·67-s + 8·72-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.755·7-s − 1.41·8-s − 2/3·9-s − 1.10·13-s − 1.06·14-s + 5/4·16-s + 0.942·18-s − 25-s + 1.56·26-s + 1.13·28-s − 2.22·29-s − 1.06·32-s − 36-s + 0.657·37-s − 3.50·47-s + 3/7·49-s + 1.41·50-s − 1.66·52-s − 1.06·56-s + 3.15·58-s + 2.04·61-s − 0.503·63-s + 7/8·64-s − 0.977·67-s + 0.942·72-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(828100\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{828100} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 828100,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.4869251355$
$L(\frac12)$  $\approx$  $0.4869251355$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−8.149115589354817244986775905302, −7.921942048273572677150608948373, −7.57571100088867902110310233811, −6.96023036898751402073285166387, −6.75497369172733229312582987288, −6.00728441107086114890911949727, −5.57928681742950427486583645839, −5.23472501559197949370067568971, −4.58610841451955891029930986059, −3.91813300576011100108011679413, −3.27110855367138212573533050585, −2.67579223441987319029898681629, −1.94147475247196906907387077179, −1.67889512896363126345389610058, −0.40214508136838586607328490895, 0.40214508136838586607328490895, 1.67889512896363126345389610058, 1.94147475247196906907387077179, 2.67579223441987319029898681629, 3.27110855367138212573533050585, 3.91813300576011100108011679413, 4.58610841451955891029930986059, 5.23472501559197949370067568971, 5.57928681742950427486583645839, 6.00728441107086114890911949727, 6.75497369172733229312582987288, 6.96023036898751402073285166387, 7.57571100088867902110310233811, 7.921942048273572677150608948373, 8.149115589354817244986775905302

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