Properties

Degree 4
Conductor $ 2^{5} \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 + 4-s − 8-s − 5·9-s − 8·13-s + 16-s − 6·17-s + 5·18-s + 8·26-s − 32-s + 6·34-s − 5·36-s + 4·37-s − 6·41-s − 10·49-s − 8·52-s + 12·53-s + 4·61-s + 64-s − 6·68-s + 5·72-s + 22·73-s − 4·74-s + 16·81-s + 6·82-s + 30·89-s + 4·97-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 5/3·9-s − 2.21·13-s + 1/4·16-s − 1.45·17-s + 1.17·18-s + 1.56·26-s − 0.176·32-s + 1.02·34-s − 5/6·36-s + 0.657·37-s − 0.937·41-s − 1.42·49-s − 1.10·52-s + 1.64·53-s + 0.512·61-s + 1/8·64-s − 0.727·68-s + 0.589·72-s + 2.57·73-s − 0.464·74-s + 16/9·81-s + 0.662·82-s + 3.17·89-s + 0.406·97-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 20000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 20000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(20000\)    =    \(2^{5} \cdot 5^{4}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{20000} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 20000,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
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$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−10.81431448769284119152476376034, −9.919702445885858877180142258118, −9.447991916742081591691197868932, −9.107735829970857706936675027245, −8.316812877075225116243141130520, −8.082389583115066239879169681269, −7.29853134793764088505009214313, −6.71237324686107378754912317869, −6.21926731108516302667159437545, −5.20592148788576143134616479765, −4.98537291304699817306164733550, −3.80421461859967166401405341307, −2.65155040417235216240951299295, −2.32235507105178670692643054585, 0, 2.32235507105178670692643054585, 2.65155040417235216240951299295, 3.80421461859967166401405341307, 4.98537291304699817306164733550, 5.20592148788576143134616479765, 6.21926731108516302667159437545, 6.71237324686107378754912317869, 7.29853134793764088505009214313, 8.082389583115066239879169681269, 8.316812877075225116243141130520, 9.107735829970857706936675027245, 9.447991916742081591691197868932, 9.919702445885858877180142258118, 10.81431448769284119152476376034

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