Properties

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

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 6·5-s − 14·7-s + 2·9-s + 20·11-s + 18·13-s − 12·15-s + 2·17-s − 28·21-s − 46·23-s + 11·25-s + 18·27-s − 28·31-s + 40·33-s + 84·35-s + 66·37-s + 36·39-s − 28·41-s − 30·43-s − 12·45-s − 78·47-s + 98·49-s + 4·51-s − 14·53-s − 120·55-s + 84·61-s − 28·63-s + ⋯
L(s)  = 1  + 2/3·3-s − 6/5·5-s − 2·7-s + 2/9·9-s + 1.81·11-s + 1.38·13-s − 4/5·15-s + 2/17·17-s − 4/3·21-s − 2·23-s + 0.439·25-s + 2/3·27-s − 0.903·31-s + 1.21·33-s + 12/5·35-s + 1.78·37-s + 0.923·39-s − 0.682·41-s − 0.697·43-s − 0.266·45-s − 1.65·47-s + 2·49-s + 4/51·51-s − 0.264·53-s − 2.18·55-s + 1.37·61-s − 4/9·63-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(400\)    =    \(2^{4} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{20} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 400,\ (\ :1, 1),\ 1)$
$L(\frac{3}{2})$  $\approx$  $0.733358$
$L(\frac12)$  $\approx$  $0.733358$
$L(2)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5\}$, \(F_p\) is a polynomial of degree 4. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
5$C_2$ \( 1 + 6 T + p^{2} T^{2} \)
good3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} \)
7$C_1$$\times$$C_2$ \( ( 1 + p T )^{2}( 1 + p^{2} T^{2} ) \)
11$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 18 T + 162 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} \)
17$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} \)
19$C_2^2$ \( 1 - 658 T^{2} + p^{4} T^{4} \)
23$C_1$$\times$$C_2$ \( ( 1 + p T )^{2}( 1 + p^{2} T^{2} ) \)
29$C_2^2$ \( 1 - 1618 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 + 14 T + p^{2} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 66 T + 2178 T^{2} - 66 p^{2} T^{3} + p^{4} T^{4} \)
41$C_2$ \( ( 1 + 14 T + p^{2} T^{2} )^{2} \)
43$C_2^2$ \( 1 + 30 T + 450 T^{2} + 30 p^{2} T^{3} + p^{4} T^{4} \)
47$C_2^2$ \( 1 + 78 T + 3042 T^{2} + 78 p^{2} T^{3} + p^{4} T^{4} \)
53$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 3826 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 - 42 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} \)
71$C_2$ \( ( 1 - 98 T + p^{2} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 98 T + 4802 T^{2} - 98 p^{2} T^{3} + p^{4} T^{4} \)
79$C_2^2$ \( 1 - 3266 T^{2} + p^{4} T^{4} \)
83$C_2^2$ \( 1 + 126 T + 7938 T^{2} + 126 p^{2} T^{3} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 3298 T^{2} + p^{4} T^{4} \)
97$C_2^2$ \( 1 - 66 T + 2178 T^{2} - 66 p^{2} T^{3} + p^{4} T^{4} \)
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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

−18.35124538472176511757358714269, −18.34748371195793641274209620802, −16.79087467384288582269870954105, −16.54259600717954489917133610160, −15.77141160655571916844795700460, −15.61439987312263760175240269667, −14.50339807607041166725982042675, −14.10552965086306807216091377726, −13.13258781276970670552049700998, −12.68128665727435992573424581356, −11.78936999005442889405940598420, −11.32836035526420098193819171841, −10.02152727589583107599767457175, −9.494964812819170688883265412784, −8.664380106294927574380010983587, −7.931175664981236379641946395835, −6.64188074165720645380338848513, −6.27462099292704690001749659643, −3.85769210437284222679930271984, −3.58149732306835412927525427068, 3.58149732306835412927525427068, 3.85769210437284222679930271984, 6.27462099292704690001749659643, 6.64188074165720645380338848513, 7.931175664981236379641946395835, 8.664380106294927574380010983587, 9.494964812819170688883265412784, 10.02152727589583107599767457175, 11.32836035526420098193819171841, 11.78936999005442889405940598420, 12.68128665727435992573424581356, 13.13258781276970670552049700998, 14.10552965086306807216091377726, 14.50339807607041166725982042675, 15.61439987312263760175240269667, 15.77141160655571916844795700460, 16.54259600717954489917133610160, 16.79087467384288582269870954105, 18.34748371195793641274209620802, 18.35124538472176511757358714269

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