Properties

Label 4-1920e2-1.1-c1e2-0-7
Degree $4$
Conductor $3686400$
Sign $1$
Analytic cond. $235.048$
Root an. cond. $3.91551$
Motivic weight $1$
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  − 2·3-s − 2·5-s − 6·7-s + 3·9-s − 2·11-s + 4·15-s + 2·17-s − 2·19-s + 12·21-s − 10·23-s − 25-s − 4·27-s + 10·29-s + 4·33-s + 12·35-s − 6·45-s + 6·47-s + 18·49-s − 4·51-s − 12·53-s + 4·55-s + 4·57-s + 10·59-s + 10·61-s − 18·63-s + 20·69-s + 32·71-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s − 2.26·7-s + 9-s − 0.603·11-s + 1.03·15-s + 0.485·17-s − 0.458·19-s + 2.61·21-s − 2.08·23-s − 1/5·25-s − 0.769·27-s + 1.85·29-s + 0.696·33-s + 2.02·35-s − 0.894·45-s + 0.875·47-s + 18/7·49-s − 0.560·51-s − 1.64·53-s + 0.539·55-s + 0.529·57-s + 1.30·59-s + 1.28·61-s − 2.26·63-s + 2.40·69-s + 3.79·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3686400\)    =    \(2^{14} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(235.048\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3686400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6462890913\)
\(L(\frac12)\) \(\approx\) \(0.6462890913\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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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

−9.447282259058496656107617026910, −9.354203636642218741511916528631, −8.399645294089345298499252225425, −8.142974017599299862871853946544, −7.910456247741476828874081328254, −7.39422794161593631671613939352, −6.82010196418059894982749939170, −6.52696134348317964042067970187, −6.19038584307397560050966699483, −6.15533985073289964670111177010, −5.20183012970687026623140419515, −5.19559558342683848042590859183, −4.50081432370701926876020645714, −3.81532789204667205157079375202, −3.67089442705599839032332503992, −3.35725872357999436390768267473, −2.33944485409719431541105213992, −2.23646684367379428658000953639, −0.69982741033426648465656315732, −0.51235093034275550841638650954, 0.51235093034275550841638650954, 0.69982741033426648465656315732, 2.23646684367379428658000953639, 2.33944485409719431541105213992, 3.35725872357999436390768267473, 3.67089442705599839032332503992, 3.81532789204667205157079375202, 4.50081432370701926876020645714, 5.19559558342683848042590859183, 5.20183012970687026623140419515, 6.15533985073289964670111177010, 6.19038584307397560050966699483, 6.52696134348317964042067970187, 6.82010196418059894982749939170, 7.39422794161593631671613939352, 7.910456247741476828874081328254, 8.142974017599299862871853946544, 8.399645294089345298499252225425, 9.354203636642218741511916528631, 9.447282259058496656107617026910

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