Properties

Label 4-240e2-1.1-c9e2-0-2
Degree $4$
Conductor $57600$
Sign $1$
Analytic cond. $15279.0$
Root an. cond. $11.1179$
Motivic weight $9$
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  − 162·3-s − 1.25e3·5-s + 1.18e4·7-s + 1.96e4·9-s − 3.54e4·11-s + 1.43e5·13-s + 2.02e5·15-s + 3.85e5·17-s + 4.03e5·19-s − 1.92e6·21-s − 2.23e5·23-s + 1.17e6·25-s − 2.12e6·27-s − 7.45e4·29-s + 5.02e6·31-s + 5.74e6·33-s − 1.48e7·35-s + 5.37e6·37-s − 2.32e7·39-s + 1.42e7·41-s − 2.77e7·43-s − 2.46e7·45-s − 9.59e7·47-s + 2.87e7·49-s − 6.23e7·51-s − 6.43e7·53-s + 4.43e7·55-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s + 1.86·7-s + 9-s − 0.730·11-s + 1.39·13-s + 1.03·15-s + 1.11·17-s + 0.709·19-s − 2.15·21-s − 0.166·23-s + 3/5·25-s − 0.769·27-s − 0.0195·29-s + 0.977·31-s + 0.843·33-s − 1.67·35-s + 0.471·37-s − 1.61·39-s + 0.785·41-s − 1.23·43-s − 0.894·45-s − 2.86·47-s + 0.711·49-s − 1.29·51-s − 1.11·53-s + 0.653·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(57600\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(15279.0\)
Root analytic conductor: \(11.1179\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 57600,\ (\ :9/2, 9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(3.017268242\)
\(L(\frac12)\) \(\approx\) \(3.017268242\)
\(L(\frac{11}{2})\) 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$C_1$ \( ( 1 + p^{4} T )^{2} \)
5$C_1$ \( ( 1 + p^{4} T )^{2} \)
good7$D_{4}$ \( 1 - 1696 p T + 327218 p^{3} T^{2} - 1696 p^{10} T^{3} + p^{18} T^{4} \)
11$D_{4}$ \( 1 + 35488 T + 526013014 T^{2} + 35488 p^{9} T^{3} + p^{18} T^{4} \)
13$D_{4}$ \( 1 - 11052 p T + 24105149134 T^{2} - 11052 p^{10} T^{3} + p^{18} T^{4} \)
17$D_{4}$ \( 1 - 385156 T + 268539949078 T^{2} - 385156 p^{9} T^{3} + p^{18} T^{4} \)
19$D_{4}$ \( 1 - 403296 T + 684929514838 T^{2} - 403296 p^{9} T^{3} + p^{18} T^{4} \)
23$D_{4}$ \( 1 + 223704 T - 1375273107794 T^{2} + 223704 p^{9} T^{3} + p^{18} T^{4} \)
29$D_{4}$ \( 1 + 74572 T + 28833430018078 T^{2} + 74572 p^{9} T^{3} + p^{18} T^{4} \)
31$D_{4}$ \( 1 - 5027128 T + 52415931233342 T^{2} - 5027128 p^{9} T^{3} + p^{18} T^{4} \)
37$D_{4}$ \( 1 - 5373628 T + 231061724951934 T^{2} - 5373628 p^{9} T^{3} + p^{18} T^{4} \)
41$D_{4}$ \( 1 - 14211332 T + 443988635955862 T^{2} - 14211332 p^{9} T^{3} + p^{18} T^{4} \)
43$D_{4}$ \( 1 + 27748920 T + 1170232974699430 T^{2} + 27748920 p^{9} T^{3} + p^{18} T^{4} \)
47$D_{4}$ \( 1 + 95966440 T + 4320659216802910 T^{2} + 95966440 p^{9} T^{3} + p^{18} T^{4} \)
53$D_{4}$ \( 1 + 64305596 T + 6611083028543086 T^{2} + 64305596 p^{9} T^{3} + p^{18} T^{4} \)
59$D_{4}$ \( 1 + 187863136 T + 23071633420288438 T^{2} + 187863136 p^{9} T^{3} + p^{18} T^{4} \)
61$D_{4}$ \( 1 - 154080060 T + 23302683905802238 T^{2} - 154080060 p^{9} T^{3} + p^{18} T^{4} \)
67$D_{4}$ \( 1 + 33592376 T - 10819815556424362 T^{2} + 33592376 p^{9} T^{3} + p^{18} T^{4} \)
71$D_{4}$ \( 1 - 228270976 T + 45777616900481806 T^{2} - 228270976 p^{9} T^{3} + p^{18} T^{4} \)
73$D_{4}$ \( 1 + 33122316 T + 68371107952007926 T^{2} + 33122316 p^{9} T^{3} + p^{18} T^{4} \)
79$D_{4}$ \( 1 - 932406760 T + 453226630902929438 T^{2} - 932406760 p^{9} T^{3} + p^{18} T^{4} \)
83$D_{4}$ \( 1 + 207040152 T + 372783310330485238 T^{2} + 207040152 p^{9} T^{3} + p^{18} T^{4} \)
89$D_{4}$ \( 1 - 2522676 p T + 610925899926766678 T^{2} - 2522676 p^{10} T^{3} + p^{18} T^{4} \)
97$D_{4}$ \( 1 - 387134596 T - 734969029248610362 T^{2} - 387134596 p^{9} T^{3} + p^{18} T^{4} \)
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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

−10.82158894371469777128875876856, −10.66683304307045214018557953336, −9.779814413332066795473311667615, −9.552242086121137198211787286300, −8.493980629516209188301480561296, −8.104019346006165527132416540627, −7.929227391751733411359698196316, −7.55440567781701985667779926941, −6.56801355279333389309555021872, −6.39788060244603872973061895633, −5.42143717186757852901116049022, −5.31269352421356884224390810981, −4.50654095703370565632822371934, −4.50286353104763628960535066187, −3.32331781717757707483685028762, −3.23076233337456084916915390970, −1.80887010508796401555275439371, −1.59687161945713009187816229021, −0.817945586224562244081279131880, −0.53087056877413037517227093137, 0.53087056877413037517227093137, 0.817945586224562244081279131880, 1.59687161945713009187816229021, 1.80887010508796401555275439371, 3.23076233337456084916915390970, 3.32331781717757707483685028762, 4.50286353104763628960535066187, 4.50654095703370565632822371934, 5.31269352421356884224390810981, 5.42143717186757852901116049022, 6.39788060244603872973061895633, 6.56801355279333389309555021872, 7.55440567781701985667779926941, 7.929227391751733411359698196316, 8.104019346006165527132416540627, 8.493980629516209188301480561296, 9.552242086121137198211787286300, 9.779814413332066795473311667615, 10.66683304307045214018557953336, 10.82158894371469777128875876856

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