Properties

Label 4-240e2-1.1-c1e2-0-12
Degree $4$
Conductor $57600$
Sign $1$
Analytic cond. $3.67262$
Root an. cond. $1.38434$
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  + 2·2-s + 2·4-s − 4·5-s + 6·7-s − 9-s − 8·10-s − 2·11-s + 12·14-s − 4·16-s + 2·17-s − 2·18-s + 2·19-s − 8·20-s − 4·22-s + 10·23-s + 11·25-s + 12·28-s + 10·29-s − 8·32-s + 4·34-s − 24·35-s − 2·36-s − 8·37-s + 4·38-s + 20·43-s − 4·44-s + 4·45-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 1.78·5-s + 2.26·7-s − 1/3·9-s − 2.52·10-s − 0.603·11-s + 3.20·14-s − 16-s + 0.485·17-s − 0.471·18-s + 0.458·19-s − 1.78·20-s − 0.852·22-s + 2.08·23-s + 11/5·25-s + 2.26·28-s + 1.85·29-s − 1.41·32-s + 0.685·34-s − 4.05·35-s − 1/3·36-s − 1.31·37-s + 0.648·38-s + 3.04·43-s − 0.603·44-s + 0.596·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(57600\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(3.67262\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 57600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.585156365\)
\(L(\frac12)\) \(\approx\) \(2.585156365\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
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$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
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$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
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

−12.30277743047256296942628628259, −11.87357146532187374300543635587, −11.73724611591079243053223828471, −10.89287960823377958795955958260, −10.88017615680844132651720903592, −10.55309467211283921745012126239, −9.108514123843448091102343693883, −8.884238322719505538202442732131, −8.348937428797636956252154513379, −7.71417380915102631851941001348, −7.49748146876464135634493508786, −7.01977533667052365082383289633, −6.07166423297227469345000542907, −5.28153495796231704831833325442, −5.04355076448595325183753732849, −4.33724659674323956503132266384, −4.27665481194036235980921539372, −3.03293358385596068798404509070, −2.83842233717376064264819265093, −1.22742543153505009537701272642, 1.22742543153505009537701272642, 2.83842233717376064264819265093, 3.03293358385596068798404509070, 4.27665481194036235980921539372, 4.33724659674323956503132266384, 5.04355076448595325183753732849, 5.28153495796231704831833325442, 6.07166423297227469345000542907, 7.01977533667052365082383289633, 7.49748146876464135634493508786, 7.71417380915102631851941001348, 8.348937428797636956252154513379, 8.884238322719505538202442732131, 9.108514123843448091102343693883, 10.55309467211283921745012126239, 10.88017615680844132651720903592, 10.89287960823377958795955958260, 11.73724611591079243053223828471, 11.87357146532187374300543635587, 12.30277743047256296942628628259

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