Properties

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

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2·3-s + 4-s − 2·6-s − 8-s − 3·9-s + 2·12-s − 8·13-s + 16-s + 3·18-s − 2·24-s + 8·26-s − 14·27-s + 4·31-s − 32-s − 3·36-s + 4·37-s − 16·39-s − 6·41-s − 8·43-s + 2·48-s − 10·49-s − 8·52-s + 12·53-s + 14·54-s − 4·62-s + 64-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.353·8-s − 9-s + 0.577·12-s − 2.21·13-s + 1/4·16-s + 0.707·18-s − 0.408·24-s + 1.56·26-s − 2.69·27-s + 0.718·31-s − 0.176·32-s − 1/2·36-s + 0.657·37-s − 2.56·39-s − 0.937·41-s − 1.21·43-s + 0.288·48-s − 1.42·49-s − 1.10·52-s + 1.64·53-s + 1.90·54-s − 0.508·62-s + 1/8·64-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  =  1
Selberg data  =  $(4,\ 80000,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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\[\begin{aligned} L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.447991916742081591691197868932, −8.984879988992018021562140214035, −8.316812877075225116243141130520, −8.305108042202074861058072254432, −7.50685211643047093775514357787, −7.29853134793764088505009214313, −6.51600833722009744085233889545, −5.92775363035311842870660338443, −5.20592148788576143134616479765, −4.74379980896905468893340943589, −3.73692800680606925562186179410, −2.93297962179085959554069000009, −2.65155040417235216240951299295, −1.90088015093802268091220378336, 0, 1.90088015093802268091220378336, 2.65155040417235216240951299295, 2.93297962179085959554069000009, 3.73692800680606925562186179410, 4.74379980896905468893340943589, 5.20592148788576143134616479765, 5.92775363035311842870660338443, 6.51600833722009744085233889545, 7.29853134793764088505009214313, 7.50685211643047093775514357787, 8.305108042202074861058072254432, 8.316812877075225116243141130520, 8.984879988992018021562140214035, 9.447991916742081591691197868932

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