Properties

Label 4-160e2-1.1-c3e2-0-1
Degree $4$
Conductor $25600$
Sign $1$
Analytic cond. $89.1193$
Root an. cond. $3.07250$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 22·5-s + 54·9-s + 359·25-s − 260·29-s + 460·41-s − 1.18e3·45-s + 686·49-s − 1.66e3·61-s + 2.18e3·81-s − 3.34e3·89-s + 1.19e3·101-s + 3.49e3·109-s − 2.66e3·121-s − 5.14e3·125-s + 127-s + 131-s + 137-s + 139-s + 5.72e3·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4.07e3·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 1.96·5-s + 2·9-s + 2.87·25-s − 1.66·29-s + 1.75·41-s − 3.93·45-s + 2·49-s − 3.48·61-s + 3·81-s − 3.97·89-s + 1.17·101-s + 3.06·109-s − 2·121-s − 3.68·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 3.27·145-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 1.85·169-s + 0.000439·173-s + 0.000417·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(25600\)    =    \(2^{10} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(89.1193\)
Root analytic conductor: \(3.07250\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 25600,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.472544994\)
\(L(\frac12)\) \(\approx\) \(1.472544994\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + 22 T + p^{3} T^{2} \)
good3$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
7$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
11$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 18 T + p^{3} T^{2} )( 1 + 18 T + p^{3} T^{2} ) \)
17$C_2$ \( ( 1 - 94 T + p^{3} T^{2} )( 1 + 94 T + p^{3} T^{2} ) \)
19$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
23$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
29$C_2$ \( ( 1 + 130 T + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 214 T + p^{3} T^{2} )( 1 + 214 T + p^{3} T^{2} ) \)
41$C_2$ \( ( 1 - 230 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
47$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
53$C_2$ \( ( 1 - 518 T + p^{3} T^{2} )( 1 + 518 T + p^{3} T^{2} ) \)
59$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 830 T + p^{3} T^{2} )^{2} \)
67$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
71$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 1098 T + p^{3} T^{2} )( 1 + 1098 T + p^{3} T^{2} ) \)
79$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
83$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
89$C_2$ \( ( 1 + 1670 T + p^{3} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 594 T + p^{3} T^{2} )( 1 + 594 T + p^{3} T^{2} ) \)
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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

−12.52933640257135710953824103606, −12.33438326636382023820350981237, −11.72079607174343008749620411088, −11.13828778032039690429166899346, −10.75882802284608102179476389221, −10.32965164092662315422138116212, −9.539483853163395076468002903218, −9.135796436023538109969097023523, −8.498110829954472867085482213523, −7.72137854563861520887544976189, −7.38132275662760033118377177323, −7.27035454540997435946165121041, −6.42903827211856340155560119709, −5.56107479388189344856367190790, −4.58320432736613966477524676957, −4.24929507049613099274517542243, −3.81647198083376130530739013199, −2.94377924312846552352482580074, −1.62138251422118657619404990248, −0.61113760722332229735968895935, 0.61113760722332229735968895935, 1.62138251422118657619404990248, 2.94377924312846552352482580074, 3.81647198083376130530739013199, 4.24929507049613099274517542243, 4.58320432736613966477524676957, 5.56107479388189344856367190790, 6.42903827211856340155560119709, 7.27035454540997435946165121041, 7.38132275662760033118377177323, 7.72137854563861520887544976189, 8.498110829954472867085482213523, 9.135796436023538109969097023523, 9.539483853163395076468002903218, 10.32965164092662315422138116212, 10.75882802284608102179476389221, 11.13828778032039690429166899346, 11.72079607174343008749620411088, 12.33438326636382023820350981237, 12.52933640257135710953824103606

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