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Properties

Label 4-40e4-1.1-c0e2-0-1
Degree $4$
Conductor $2560000$
Sign $1$
Analytic cond. $0.637608$
Root an. cond. $0.893590$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·19-s + 4·41-s + 4·59-s − 81-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  − 4·19-s + 4·41-s + 4·59-s − 81-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2560000\)    =    \(2^{12} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(0.637608\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2560000,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9549248127\)
\(L(\frac12)\) \(\approx\) \(0.9549248127\)
\(L(1)\) 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 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 + T^{4} \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_1$ \( ( 1 + T )^{4} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_1$ \( ( 1 - T )^{4} \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_2^2$ \( 1 + T^{4} \)
59$C_1$ \( ( 1 - T )^{4} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
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

−9.852823589809812008632608163208, −9.247111770411681965366853272028, −8.997188209433688262094213228938, −8.643143316780750637366533237565, −8.100557276558661501375151136729, −8.087037699184022222327052723812, −7.39500675848415527813322902401, −6.82127411470994223871723853581, −6.68337555070165749539635926378, −6.19341001085254545499462398235, −5.65673538214919461925099154589, −5.55181477251548754789703190542, −4.57772335230375983126653523610, −4.22308004814163926392777277835, −4.19604982427584580125755303700, −3.52001274082573167693286635440, −2.51676695330962227028664132581, −2.44073975174804887243873314855, −1.87141801462053587037809439727, −0.76618312973640331473668257967, 0.76618312973640331473668257967, 1.87141801462053587037809439727, 2.44073975174804887243873314855, 2.51676695330962227028664132581, 3.52001274082573167693286635440, 4.19604982427584580125755303700, 4.22308004814163926392777277835, 4.57772335230375983126653523610, 5.55181477251548754789703190542, 5.65673538214919461925099154589, 6.19341001085254545499462398235, 6.68337555070165749539635926378, 6.82127411470994223871723853581, 7.39500675848415527813322902401, 8.087037699184022222327052723812, 8.100557276558661501375151136729, 8.643143316780750637366533237565, 8.997188209433688262094213228938, 9.247111770411681965366853272028, 9.852823589809812008632608163208

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