Properties

Degree $4$
Conductor $101124$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 2·7-s − 9-s + 10·11-s + 4·13-s + 16-s − 6·25-s − 2·28-s + 10·29-s + 36-s − 20·37-s + 12·43-s − 10·44-s + 12·47-s − 11·49-s − 4·52-s + 14·53-s − 24·59-s − 2·63-s − 64-s + 20·77-s + 81-s − 28·89-s + 8·91-s + 10·97-s − 10·99-s + 6·100-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.755·7-s − 1/3·9-s + 3.01·11-s + 1.10·13-s + 1/4·16-s − 6/5·25-s − 0.377·28-s + 1.85·29-s + 1/6·36-s − 3.28·37-s + 1.82·43-s − 1.50·44-s + 1.75·47-s − 1.57·49-s − 0.554·52-s + 1.92·53-s − 3.12·59-s − 0.251·63-s − 1/8·64-s + 2.27·77-s + 1/9·81-s − 2.96·89-s + 0.838·91-s + 1.01·97-s − 1.00·99-s + 3/5·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101124 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101124 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(101124\)    =    \(2^{2} \cdot 3^{2} \cdot 53^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{101124} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 101124,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.835915627\)
\(L(\frac12)\) \(\approx\) \(1.835915627\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
53$C_2$ \( 1 - 14 T + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 47 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 93 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.355928540933491740730661229417, −8.890177834676785330253663348662, −8.841471073307060467201279866998, −8.239704885576410988410465565015, −7.68077436283308438393520759525, −6.74422658776545809366699068131, −6.73866635300919625217018803122, −5.93426590030487536300365765583, −5.57152941841203308210347575606, −4.68040532212025185086542620358, −4.02812088271024247271571331199, −3.90878468206067804936956687278, −3.04304548084901527146420934052, −1.69717907455138203769571317989, −1.20947970709776881780644882711, 1.20947970709776881780644882711, 1.69717907455138203769571317989, 3.04304548084901527146420934052, 3.90878468206067804936956687278, 4.02812088271024247271571331199, 4.68040532212025185086542620358, 5.57152941841203308210347575606, 5.93426590030487536300365765583, 6.73866635300919625217018803122, 6.74422658776545809366699068131, 7.68077436283308438393520759525, 8.239704885576410988410465565015, 8.841471073307060467201279866998, 8.890177834676785330253663348662, 9.355928540933491740730661229417

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