Properties

Degree 4
Conductor $ 2^{7} \cdot 5^{4} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 4·7-s − 8-s − 5·9-s − 4·14-s + 16-s − 6·17-s + 5·18-s + 12·23-s + 4·28-s + 4·31-s − 32-s + 6·34-s − 5·36-s − 6·41-s − 12·46-s + 24·47-s − 2·49-s − 4·56-s − 4·62-s − 20·63-s + 64-s − 6·68-s + 24·71-s + 5·72-s + 22·73-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s − 5/3·9-s − 1.06·14-s + 1/4·16-s − 1.45·17-s + 1.17·18-s + 2.50·23-s + 0.755·28-s + 0.718·31-s − 0.176·32-s + 1.02·34-s − 5/6·36-s − 0.937·41-s − 1.76·46-s + 3.50·47-s − 2/7·49-s − 0.534·56-s − 0.508·62-s − 2.51·63-s + 1/8·64-s − 0.727·68-s + 2.84·71-s + 0.589·72-s + 2.57·73-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 80000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 80000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(80000\)    =    \(2^{7} \cdot 5^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{80000} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 80000,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.078912628$
$L(\frac12)$  $\approx$  $1.078912628$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( 1 + T \)
5 \( 1 \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.447991916742081591691197868932, −9.108795455289480583011416986557, −8.746535765801052966997753191886, −8.316812877075225116243141130520, −7.945377299913127727532994829665, −7.29853134793764088505009214313, −6.69992826322664140620538681665, −6.29316757927509488643773547834, −5.29439943415338779506744522168, −5.20592148788576143134616479765, −4.48531193864795315694041050402, −3.53176922528875095704922238510, −2.65155040417235216240951299295, −2.18562128408643501857872580519, −0.919526999450248053192487619525, 0.919526999450248053192487619525, 2.18562128408643501857872580519, 2.65155040417235216240951299295, 3.53176922528875095704922238510, 4.48531193864795315694041050402, 5.20592148788576143134616479765, 5.29439943415338779506744522168, 6.29316757927509488643773547834, 6.69992826322664140620538681665, 7.29853134793764088505009214313, 7.945377299913127727532994829665, 8.316812877075225116243141130520, 8.746535765801052966997753191886, 9.108795455289480583011416986557, 9.447991916742081591691197868932

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