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Properties

Label 4-40e4-1.1-c0e2-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(1.340204092\)
\(L(\frac12)\) \(\approx\) \(1.340204092\)
\(L(1)\) 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

−9.515687293314207306868340307613, −9.468427319567913503046470733907, −9.183138683078763071342271146553, −8.790894359296079559353858828535, −7.88094944477844520739144757182, −7.81564094393609941575406058867, −7.44055094425025743718440609229, −7.28964249665577355792829972999, −6.52495618244382662242515758302, −6.09694848561466230782235868140, −5.60393317637493827206774621348, −5.49332326171988530287327471354, −4.66999188098055674610608618402, −4.62688692176229748117190839873, −3.73177372241674350216858593412, −3.43861809253370320929516180893, −2.71884513263521141784660689479, −2.63670773378722526500976379897, −1.34023894076697230217001540114, −1.14368493445802448259459956203, 1.14368493445802448259459956203, 1.34023894076697230217001540114, 2.63670773378722526500976379897, 2.71884513263521141784660689479, 3.43861809253370320929516180893, 3.73177372241674350216858593412, 4.62688692176229748117190839873, 4.66999188098055674610608618402, 5.49332326171988530287327471354, 5.60393317637493827206774621348, 6.09694848561466230782235868140, 6.52495618244382662242515758302, 7.28964249665577355792829972999, 7.44055094425025743718440609229, 7.81564094393609941575406058867, 7.88094944477844520739144757182, 8.790894359296079559353858828535, 9.183138683078763071342271146553, 9.468427319567913503046470733907, 9.515687293314207306868340307613

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