↑

Properties

Label 4-665e2-1.1-c0e2-0-2
Degree $4$
Conductor $442225$
Sign $1$
Analytic cond. $0.110143$
Root an. cond. $0.576088$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 16-s + 2·17-s − 2·19-s − 2·23-s + 3·25-s + 2·43-s − 2·47-s − 49-s − 2·73-s − 2·80-s − 81-s − 2·83-s + 4·85-s − 4·95-s − 4·115-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 2·5-s − 16-s + 2·17-s − 2·19-s − 2·23-s + 3·25-s + 2·43-s − 2·47-s − 49-s − 2·73-s − 2·80-s − 81-s − 2·83-s + 4·85-s − 4·95-s − 4·115-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 442225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(442225\)    =    \(5^{2} \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.110143\)
Root analytic conductor: \(0.576088\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 442225,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.147099242\)
\(L(\frac12)\) \(\approx\) \(1.147099242\)
\(L(1)\) 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)$
bad5$C_1$ \( ( 1 - T )^{2} \)
7$C_2$ \( 1 + T^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + T^{4} \)
3$C_2^2$ \( 1 + T^{4} \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
23$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
47$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
53$C_2^2$ \( 1 + T^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_2^2$ \( 1 + T^{4} \)
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

−10.72615927969041585731995588934, −10.32918905788299336334882037230, −9.988376377710077954788629609076, −9.924446417475096846171340070518, −9.138144032372668070630783235816, −9.030199195551708318378212193397, −8.378705614004823319141208676483, −7.975188589667517137895589762436, −7.48599048248338289994439141166, −6.68978854237765795045860714910, −6.47165642711306764012317447542, −6.03041039182173119252884507528, −5.57683985062434647262916555687, −5.28188730254292165510384571542, −4.30696940431278640308083674493, −4.26811540361817901088843956979, −3.12721838566535812553793476446, −2.65469169745686763492984916862, −1.91808551249598145800203958344, −1.54435994057161357153769802834, 1.54435994057161357153769802834, 1.91808551249598145800203958344, 2.65469169745686763492984916862, 3.12721838566535812553793476446, 4.26811540361817901088843956979, 4.30696940431278640308083674493, 5.28188730254292165510384571542, 5.57683985062434647262916555687, 6.03041039182173119252884507528, 6.47165642711306764012317447542, 6.68978854237765795045860714910, 7.48599048248338289994439141166, 7.975188589667517137895589762436, 8.378705614004823319141208676483, 9.030199195551708318378212193397, 9.138144032372668070630783235816, 9.924446417475096846171340070518, 9.988376377710077954788629609076, 10.32918905788299336334882037230, 10.72615927969041585731995588934

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