Properties

Label 4-1620e2-1.1-c0e2-0-1
Degree $4$
Conductor $2624400$
Sign $1$
Analytic cond. $0.653648$
Root an. cond. $0.899158$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s + 2·11-s + 2·17-s + 2·23-s − 25-s + 2·31-s + 2·41-s + 2·49-s − 2·53-s − 2·67-s − 2·71-s + 2·73-s − 4·77-s − 2·101-s + 2·103-s + 2·107-s − 2·113-s − 4·119-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 4·161-s + ⋯
L(s)  = 1  − 2·7-s + 2·11-s + 2·17-s + 2·23-s − 25-s + 2·31-s + 2·41-s + 2·49-s − 2·53-s − 2·67-s − 2·71-s + 2·73-s − 4·77-s − 2·101-s + 2·103-s + 2·107-s − 2·113-s − 4·119-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 4·161-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2624400\)    =    \(2^{4} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.653648\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2624400,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.173674835\)
\(L(\frac12)\) \(\approx\) \(1.173674835\)
\(L(1)\) 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)$
bad2 \( 1 \)
3 \( 1 \)
5$C_2$ \( 1 + T^{2} \)
good7$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
11$C_2$ \( ( 1 - T + T^{2} )^{2} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
19$C_2^2$ \( 1 - T^{2} + T^{4} \)
23$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
29$C_2^2$ \( 1 - T^{2} + T^{4} \)
31$C_2$ \( ( 1 - T + T^{2} )^{2} \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_2$ \( ( 1 - T + T^{2} )^{2} \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
59$C_2^2$ \( 1 - T^{2} + T^{4} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
71$C_2$ \( ( 1 + T + T^{2} )^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_2^2$ \( 1 - T^{2} + T^{4} \)
97$C_2^2$ \( 1 + T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.817285018251890730737094461439, −9.429301058329989485656193080175, −8.983044352910611235262376047170, −8.982538672279099593675159929244, −8.051854261980889296934999516978, −7.889104379218247201104466059728, −7.17467683447287300439154983343, −7.06408967374952688612976649064, −6.42501420608906908551344626600, −6.19403952810484311052916518592, −5.97066965238798052319730917113, −5.43908222485347726288412255275, −4.63850815168763443828917117720, −4.41326031949971868231633647385, −3.59793560300391206916986020241, −3.48768316540448623648721292838, −2.97790725315358273594864002799, −2.55699935156323982454638098293, −1.34082642338976775364110092032, −1.03116483750808352984009725253, 1.03116483750808352984009725253, 1.34082642338976775364110092032, 2.55699935156323982454638098293, 2.97790725315358273594864002799, 3.48768316540448623648721292838, 3.59793560300391206916986020241, 4.41326031949971868231633647385, 4.63850815168763443828917117720, 5.43908222485347726288412255275, 5.97066965238798052319730917113, 6.19403952810484311052916518592, 6.42501420608906908551344626600, 7.06408967374952688612976649064, 7.17467683447287300439154983343, 7.889104379218247201104466059728, 8.051854261980889296934999516978, 8.982538672279099593675159929244, 8.983044352910611235262376047170, 9.429301058329989485656193080175, 9.817285018251890730737094461439

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