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Properties

Label 4-1476e2-1.1-c0e2-0-2
Degree $4$
Conductor $2178576$
Sign $1$
Analytic cond. $0.542608$
Root an. cond. $0.858265$
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-s + 2·13-s + 16-s + 2·17-s + 2·25-s − 2·29-s − 2·52-s + 2·53-s − 64-s − 2·68-s + 2·89-s − 2·97-s − 2·100-s − 2·101-s + 2·109-s − 4·113-s + 2·116-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + ⋯
L(s)  = 1  − 4-s + 2·13-s + 16-s + 2·17-s + 2·25-s − 2·29-s − 2·52-s + 2·53-s − 64-s − 2·68-s + 2·89-s − 2·97-s − 2·100-s − 2·101-s + 2·109-s − 4·113-s + 2·116-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2178576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2178576\)    =    \(2^{4} \cdot 3^{4} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(0.542608\)
Root analytic conductor: \(0.858265\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2178576,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.099784674\)
\(L(\frac12)\) \(\approx\) \(1.099784674\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
41$C_2$ \( 1 + T^{2} \)
good5$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_2^2$ \( 1 + T^{4} \)
13$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
17$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_2^2$ \( 1 + T^{4} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
97$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
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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

−9.789623596184707533682321321909, −9.402225675549973328105393720114, −9.121278935299798544677441933676, −8.687781817255862044710192032699, −8.310957589008608144979482603751, −8.118089993795543680943291606277, −7.44336070502121015735224256046, −7.25553868160207520894059206462, −6.60013275495954908214023408458, −6.10142290643296686356616402694, −5.73399058890698521265318056840, −5.22230581657981943464323655956, −5.18999391922972354499396520961, −4.25306803581045122997010486618, −3.93586924915588140063786467564, −3.45054400964581410388395363340, −3.21718072262892305912603553505, −2.38518527345071195405371370170, −1.24503092879657175657303625960, −1.15677489399047933799008887389, 1.15677489399047933799008887389, 1.24503092879657175657303625960, 2.38518527345071195405371370170, 3.21718072262892305912603553505, 3.45054400964581410388395363340, 3.93586924915588140063786467564, 4.25306803581045122997010486618, 5.18999391922972354499396520961, 5.22230581657981943464323655956, 5.73399058890698521265318056840, 6.10142290643296686356616402694, 6.60013275495954908214023408458, 7.25553868160207520894059206462, 7.44336070502121015735224256046, 8.118089993795543680943291606277, 8.310957589008608144979482603751, 8.687781817255862044710192032699, 9.121278935299798544677441933676, 9.402225675549973328105393720114, 9.789623596184707533682321321909

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