Properties

Label 4-1840e2-1.1-c1e2-0-2
Degree $4$
Conductor $3385600$
Sign $1$
Analytic cond. $215.868$
Root an. cond. $3.83307$
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  + 3-s − 2·5-s + 7-s − 9-s + 4·11-s − 3·13-s − 2·15-s + 7·17-s + 4·19-s + 21-s + 2·23-s + 3·25-s − 8·29-s + 8·31-s + 4·33-s − 2·35-s − 3·37-s − 3·39-s + 4·43-s + 2·45-s + 5·47-s − 9·49-s + 7·51-s + 13·53-s − 8·55-s + 4·57-s + 7·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 0.377·7-s − 1/3·9-s + 1.20·11-s − 0.832·13-s − 0.516·15-s + 1.69·17-s + 0.917·19-s + 0.218·21-s + 0.417·23-s + 3/5·25-s − 1.48·29-s + 1.43·31-s + 0.696·33-s − 0.338·35-s − 0.493·37-s − 0.480·39-s + 0.609·43-s + 0.298·45-s + 0.729·47-s − 9/7·49-s + 0.980·51-s + 1.78·53-s − 1.07·55-s + 0.529·57-s + 0.911·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3385600\)    =    \(2^{8} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(215.868\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3385600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.138925745\)
\(L(\frac12)\) \(\approx\) \(3.138925745\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
23$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 8 T + 61 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 3 T + 72 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 65 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 5 T + 62 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 13 T + 144 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 7 T + 126 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 10 T + 130 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 19 T + 220 T^{2} - 19 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 - 11 T + 172 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 10 T + 166 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 3 T + 130 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 - 2 T + 178 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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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

−9.271554208610847512874259367228, −9.250587453071981755757833226083, −8.410697399423667607673292328043, −8.385665759754422791383462165258, −7.88294530654859239205658542762, −7.56805301104059444714592387550, −7.08871209573266585050090630276, −6.97118841871579917196816961151, −6.23074658062119194078754155496, −5.87066376810714800627134619033, −5.21986241986231057848261535406, −5.05115189617461357764447206848, −4.51316026171119262013341036708, −3.85676555192980723489072007430, −3.45790423205781752755255006065, −3.37702457629928921206777920413, −2.52501082421246270004418908529, −2.11852769648002980913428009938, −1.14551146797535563427591780627, −0.75533077942523569476806784582, 0.75533077942523569476806784582, 1.14551146797535563427591780627, 2.11852769648002980913428009938, 2.52501082421246270004418908529, 3.37702457629928921206777920413, 3.45790423205781752755255006065, 3.85676555192980723489072007430, 4.51316026171119262013341036708, 5.05115189617461357764447206848, 5.21986241986231057848261535406, 5.87066376810714800627134619033, 6.23074658062119194078754155496, 6.97118841871579917196816961151, 7.08871209573266585050090630276, 7.56805301104059444714592387550, 7.88294530654859239205658542762, 8.385665759754422791383462165258, 8.410697399423667607673292328043, 9.250587453071981755757833226083, 9.271554208610847512874259367228

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