Properties

Degree 4
Conductor $ 2^{6} \cdot 3^{4} \cdot 5^{4} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s − 2·13-s + 2·19-s + 2·31-s − 2·43-s + 49-s + 2·61-s + 2·67-s − 4·91-s − 2·97-s + 4·103-s − 2·109-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 2·7-s − 2·13-s + 2·19-s + 2·31-s − 2·43-s + 49-s + 2·61-s + 2·67-s − 4·91-s − 2·97-s + 4·103-s − 2·109-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3240000\)    =    \(2^{6} \cdot 3^{4} \cdot 5^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{1800} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 3240000,\ (\ :0, 0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $1.513027032$
$L(\frac12)$  $\approx$  $1.513027032$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \(F_p\) is a polynomial of degree 4. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7$C_2$ \( ( 1 - T + T^{2} )^{2} \)
11$C_2^2$ \( 1 + T^{4} \)
13$C_2$ \( ( 1 + T + T^{2} )^{2} \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_2$ \( ( 1 - T + T^{2} )^{2} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_2^2$ \( 1 + T^{4} \)
31$C_2$ \( ( 1 - T + T^{2} )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 + T + T^{2} )^{2} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_2$ \( ( 1 - T + T^{2} )^{2} \)
67$C_2$ \( ( 1 - T + T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_2$ \( ( 1 + T + T^{2} )^{2} \)
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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

−9.726849853760162435945790524760, −9.481643109819209187920328009155, −8.571117989757483871185316019410, −8.516328304255330979798076045165, −8.058916373388292310969097090446, −7.77888556726321503794629606575, −7.25221067118758668698812393630, −7.15616098528516266801275248004, −6.55212368187699617938367334150, −6.05150573135710332496011069713, −5.27200937995422452314448589512, −5.15345747603836662233310407075, −4.83784542716465231509154655959, −4.62705114421934455860040065344, −3.80244267575712987763138962422, −3.37350855651480880371050693834, −2.50054492117105069112874598586, −2.43891429590937091663326239392, −1.56505875568933193688038560825, −1.04320301703188116681147743593, 1.04320301703188116681147743593, 1.56505875568933193688038560825, 2.43891429590937091663326239392, 2.50054492117105069112874598586, 3.37350855651480880371050693834, 3.80244267575712987763138962422, 4.62705114421934455860040065344, 4.83784542716465231509154655959, 5.15345747603836662233310407075, 5.27200937995422452314448589512, 6.05150573135710332496011069713, 6.55212368187699617938367334150, 7.15616098528516266801275248004, 7.25221067118758668698812393630, 7.77888556726321503794629606575, 8.058916373388292310969097090446, 8.516328304255330979798076045165, 8.571117989757483871185316019410, 9.481643109819209187920328009155, 9.726849853760162435945790524760

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