Properties

Label 4-800e2-1.1-c0e2-0-0
Degree $4$
Conductor $640000$
Sign $1$
Analytic cond. $0.159402$
Root an. cond. $0.631863$
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  − 2·3-s + 2·7-s + 2·9-s − 4·21-s − 2·23-s − 2·27-s + 2·43-s − 2·47-s + 2·49-s + 4·63-s + 2·67-s + 4·69-s + 3·81-s + 2·83-s + 4·101-s + 2·103-s − 2·107-s − 2·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s + 4·141-s − 4·147-s + 149-s + 151-s + ⋯
L(s)  = 1  − 2·3-s + 2·7-s + 2·9-s − 4·21-s − 2·23-s − 2·27-s + 2·43-s − 2·47-s + 2·49-s + 4·63-s + 2·67-s + 4·69-s + 3·81-s + 2·83-s + 4·101-s + 2·103-s − 2·107-s − 2·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s + 4·141-s − 4·147-s + 149-s + 151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(640000\)    =    \(2^{10} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(0.159402\)
Root analytic conductor: \(0.631863\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 640000,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5353003331\)
\(L(\frac12)\) \(\approx\) \(0.5353003331\)
\(L(1)\) 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_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
7$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
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_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
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

−10.75221685985421843874095155583, −10.52182806991915867942435966588, −10.03268865522979731250990617072, −9.516285885690007470604571955858, −9.099879670111464566614379913831, −8.354335030273642507666476163441, −7.987549958995614787489586635325, −7.79309233068281090636853498539, −7.23989387664129141346164886590, −6.64426959537030984101543600395, −6.05031302786525693150766574450, −5.94671590199537883366208087967, −5.35118785830813710866323432898, −4.90110252422022193942073224709, −4.68241429286250276903957052486, −4.04655376704796702448425754963, −3.53643054532470293052896772089, −2.16891100401709272297954391108, −1.91087069452013848845759771743, −0.912779280286979310418994085698, 0.912779280286979310418994085698, 1.91087069452013848845759771743, 2.16891100401709272297954391108, 3.53643054532470293052896772089, 4.04655376704796702448425754963, 4.68241429286250276903957052486, 4.90110252422022193942073224709, 5.35118785830813710866323432898, 5.94671590199537883366208087967, 6.05031302786525693150766574450, 6.64426959537030984101543600395, 7.23989387664129141346164886590, 7.79309233068281090636853498539, 7.987549958995614787489586635325, 8.354335030273642507666476163441, 9.099879670111464566614379913831, 9.516285885690007470604571955858, 10.03268865522979731250990617072, 10.52182806991915867942435966588, 10.75221685985421843874095155583

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