Properties

Degree 4
Conductor $ 2^{2} \cdot 5^{2} \cdot 43^{2} $
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  + 2·2-s + 3·4-s + 2·5-s + 2·7-s + 4·8-s − 4·9-s + 4·10-s + 4·11-s + 2·13-s + 4·14-s + 5·16-s − 8·18-s − 2·19-s + 6·20-s + 8·22-s + 4·23-s + 3·25-s + 4·26-s + 6·28-s − 6·29-s + 2·31-s + 6·32-s + 4·35-s − 12·36-s − 4·37-s − 4·38-s + 8·40-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.894·5-s + 0.755·7-s + 1.41·8-s − 4/3·9-s + 1.26·10-s + 1.20·11-s + 0.554·13-s + 1.06·14-s + 5/4·16-s − 1.88·18-s − 0.458·19-s + 1.34·20-s + 1.70·22-s + 0.834·23-s + 3/5·25-s + 0.784·26-s + 1.13·28-s − 1.11·29-s + 0.359·31-s + 1.06·32-s + 0.676·35-s − 2·36-s − 0.657·37-s − 0.648·38-s + 1.26·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 184900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(184900\)    =    \(2^{2} \cdot 5^{2} \cdot 43^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{430} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 184900,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $5.22806$
$L(\frac12)$  $\approx$  $5.22806$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;43\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
43$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$D_{4}$ \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 2 T + 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - 4 T - 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 6 T + 49 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T + 61 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 2 T + 75 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 14 T + 121 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 6 T + 125 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 10 T + 153 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 14 T + 205 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 4 T + 132 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 4 T - 44 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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

−11.30624787575493870781868277133, −11.13334187713627914064632994170, −10.81183372735846856091763662626, −10.24098341812761378480223771638, −9.351094299492888227524144359285, −9.334299586104098062957013815812, −8.510999939426121827868389744923, −8.335380592868157649115040679983, −7.54034187101934031417881441002, −7.02615058530844605478785323088, −6.31136087914550595721001935444, −6.25760926357992075675274873255, −5.47706094261517588503871830415, −5.37469575261544498847155120468, −4.48870149271777877643396807300, −4.20580724923551974763393766994, −3.19624398199023331230385161918, −3.01239367251456204280302943165, −1.95870692717463030361380630541, −1.45877734356199781399905351842, 1.45877734356199781399905351842, 1.95870692717463030361380630541, 3.01239367251456204280302943165, 3.19624398199023331230385161918, 4.20580724923551974763393766994, 4.48870149271777877643396807300, 5.37469575261544498847155120468, 5.47706094261517588503871830415, 6.25760926357992075675274873255, 6.31136087914550595721001935444, 7.02615058530844605478785323088, 7.54034187101934031417881441002, 8.335380592868157649115040679983, 8.510999939426121827868389744923, 9.334299586104098062957013815812, 9.351094299492888227524144359285, 10.24098341812761378480223771638, 10.81183372735846856091763662626, 11.13334187713627914064632994170, 11.30624787575493870781868277133

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