Properties

Degree 4
Conductor $ 2^{10} \cdot 5^{4} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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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

\( d \)  =  \(4\)
\( N \)  =  \(640000\)    =    \(2^{10} \cdot 5^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{800} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 640000,\ (\ :0, 0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.5353003331$
$L(\frac12)$  $\approx$  $0.5353003331$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5\}$, \(F_p\) is a polynomial of degree 4. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
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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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

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