Properties

Label 4-2352e2-1.1-c0e2-0-5
Degree $4$
Conductor $5531904$
Sign $1$
Analytic cond. $1.37780$
Root an. cond. $1.08342$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 19-s + 25-s − 27-s − 31-s − 37-s + 57-s + 3·67-s + 3·73-s + 75-s − 3·79-s − 81-s − 93-s − 103-s + 109-s − 111-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + ⋯
L(s)  = 1  + 3-s + 19-s + 25-s − 27-s − 31-s − 37-s + 57-s + 3·67-s + 3·73-s + 75-s − 3·79-s − 81-s − 93-s − 103-s + 109-s − 111-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5531904\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1.37780\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5531904,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.766636660\)
\(L(\frac12)\) \(\approx\) \(1.766636660\)
\(L(1)\) 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 \( 1 \)
3$C_2$ \( 1 - T + T^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 - T^{2} + T^{4} \)
11$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
17$C_2^2$ \( 1 - T^{2} + T^{4} \)
19$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
23$C_2^2$ \( 1 - T^{2} + T^{4} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
47$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
53$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
79$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2^2$ \( 1 - T^{2} + T^{4} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−9.386725719406232762259228325723, −8.829431542911691174532891929046, −8.632883593553366184180235509258, −8.285130916565853550874192986572, −7.943640985036925301273516970132, −7.44131565601255858007011060816, −7.10305047058641789067305016253, −6.85300142354819138364791964514, −6.30792394344717580039821369900, −5.79350615601559685751274934500, −5.29843358412518647895401140583, −5.16571731499131676393689109910, −4.57264677772177362081331077248, −3.82989702312932093486584324411, −3.72928945178457173015163671959, −3.11110430347519644808400840119, −2.81490086691775600320431139664, −2.15346985224634997342782038706, −1.74336759877131082053515560156, −0.876804002733165351353712806347, 0.876804002733165351353712806347, 1.74336759877131082053515560156, 2.15346985224634997342782038706, 2.81490086691775600320431139664, 3.11110430347519644808400840119, 3.72928945178457173015163671959, 3.82989702312932093486584324411, 4.57264677772177362081331077248, 5.16571731499131676393689109910, 5.29843358412518647895401140583, 5.79350615601559685751274934500, 6.30792394344717580039821369900, 6.85300142354819138364791964514, 7.10305047058641789067305016253, 7.44131565601255858007011060816, 7.943640985036925301273516970132, 8.285130916565853550874192986572, 8.632883593553366184180235509258, 8.829431542911691174532891929046, 9.386725719406232762259228325723

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