Properties

Degree 4
Conductor $ 3^{2} \cdot 23^{2} \cdot 29^{2} $
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  + 2·2-s + 2·4-s + 2·8-s − 9-s + 3·16-s − 2·18-s + 2·25-s − 2·31-s + 4·32-s − 2·36-s − 2·41-s + 2·47-s + 2·49-s + 4·50-s − 4·62-s + 4·64-s − 2·72-s − 2·73-s + 81-s − 4·82-s + 4·94-s + 4·98-s + 4·100-s − 2·101-s − 4·124-s + 127-s + 4·128-s + ⋯
L(s)  = 1  + 2·2-s + 2·4-s + 2·8-s − 9-s + 3·16-s − 2·18-s + 2·25-s − 2·31-s + 4·32-s − 2·36-s − 2·41-s + 2·47-s + 2·49-s + 4·50-s − 4·62-s + 4·64-s − 2·72-s − 2·73-s + 81-s − 4·82-s + 4·94-s + 4·98-s + 4·100-s − 2·101-s − 4·124-s + 127-s + 4·128-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4004001 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4004001 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(4004001\)    =    \(3^{2} \cdot 23^{2} \cdot 29^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{2001} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 4004001,\ (\ :0, 0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $3.739461862$
$L(\frac12)$  $\approx$  $3.739461862$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;23,\;29\}$, \(F_p\) is a polynomial of degree 4. If $p \in \{3,\;23,\;29\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_2$ \( 1 + T^{2} \)
23$C_2$ \( 1 + T^{2} \)
29$C_2$ \( 1 + T^{2} \)
good2$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
5$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
11$C_2^2$ \( 1 + T^{4} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_2^2$ \( 1 + T^{4} \)
31$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
79$C_2^2$ \( 1 + T^{4} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_2^2$ \( 1 + T^{4} \)
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

−9.448714402438412153078856361613, −9.013583199878180017521722825469, −8.818386907003194553262343213001, −8.311054210908338613961222226780, −7.909930291327359386067163393486, −7.38519636203980971489244978722, −7.01173845027374617510831424490, −6.82301674379719791647076645185, −6.08930402049363758613191143349, −5.77487939191565230315492397537, −5.46915395071510417297680507734, −5.06090099359527545749738494172, −4.84606035679535512913514622259, −4.14873997327109448197734214665, −3.85620080643315639556510783474, −3.48871735581238695571082916185, −2.76570478491314705304799496869, −2.71534346064884293610064227614, −1.80863970995744292176853033784, −1.11946198236062556720060919945, 1.11946198236062556720060919945, 1.80863970995744292176853033784, 2.71534346064884293610064227614, 2.76570478491314705304799496869, 3.48871735581238695571082916185, 3.85620080643315639556510783474, 4.14873997327109448197734214665, 4.84606035679535512913514622259, 5.06090099359527545749738494172, 5.46915395071510417297680507734, 5.77487939191565230315492397537, 6.08930402049363758613191143349, 6.82301674379719791647076645185, 7.01173845027374617510831424490, 7.38519636203980971489244978722, 7.909930291327359386067163393486, 8.311054210908338613961222226780, 8.818386907003194553262343213001, 9.013583199878180017521722825469, 9.448714402438412153078856361613

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