Properties

Label 4-52e2-1.1-c0e2-0-0
Degree $4$
Conductor $2704$
Sign $1$
Analytic cond. $0.000673474$
Root an. cond. $0.161094$
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-s − 2·5-s + 8-s − 9-s + 2·10-s − 13-s − 16-s + 17-s + 18-s + 25-s + 26-s + 29-s − 34-s + 37-s − 2·40-s + 41-s + 2·45-s − 49-s − 50-s − 2·53-s − 58-s + 61-s + 64-s + 2·65-s − 72-s − 2·73-s − 74-s + ⋯
L(s)  = 1  − 2-s − 2·5-s + 8-s − 9-s + 2·10-s − 13-s − 16-s + 17-s + 18-s + 25-s + 26-s + 29-s − 34-s + 37-s − 2·40-s + 41-s + 2·45-s − 49-s − 50-s − 2·53-s − 58-s + 61-s + 64-s + 2·65-s − 72-s − 2·73-s − 74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2704\)    =    \(2^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.000673474\)
Root analytic conductor: \(0.161094\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{52} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2704,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1055166369\)
\(L(\frac12)\) \(\approx\) \(0.1055166369\)
\(L(1)\) 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$C_2$ \( 1 + T + T^{2} \)
13$C_2$ \( 1 + T + T^{2} \)
good3$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
5$C_2$ \( ( 1 + T + T^{2} )^{2} \)
7$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
17$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
19$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
23$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
29$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
41$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
43$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_2$ \( ( 1 + T + T^{2} )^{2} \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
71$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
73$C_2$ \( ( 1 + T + T^{2} )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2$ \( ( 1 + T + T^{2} )^{2} \)
97$C_2$ \( ( 1 + T + T^{2} )^{2} \)
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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

−15.87294873793952215742661356483, −15.86901024324633578237109774276, −14.96124058887919042375486728325, −14.36669503053971576318634699798, −14.12767115902838684115562210190, −13.06765961975395203124255805669, −12.40865268975567652680377933239, −11.93591325352333454299037842375, −11.24632239324280198745131969122, −11.08546064930377023602876475562, −9.896784789924800205671186547144, −9.709118972498971638443287007627, −8.532649398844231199448686039577, −8.339948637340426939401113737946, −7.53136053749764317001096925301, −7.39766540537995423654738286953, −6.04668941269644592254494882943, −4.87537373101459305382914912093, −4.16168360291039984927090654039, −3.05212385545771777947032732532, 3.05212385545771777947032732532, 4.16168360291039984927090654039, 4.87537373101459305382914912093, 6.04668941269644592254494882943, 7.39766540537995423654738286953, 7.53136053749764317001096925301, 8.339948637340426939401113737946, 8.532649398844231199448686039577, 9.709118972498971638443287007627, 9.896784789924800205671186547144, 11.08546064930377023602876475562, 11.24632239324280198745131969122, 11.93591325352333454299037842375, 12.40865268975567652680377933239, 13.06765961975395203124255805669, 14.12767115902838684115562210190, 14.36669503053971576318634699798, 14.96124058887919042375486728325, 15.86901024324633578237109774276, 15.87294873793952215742661356483

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