Properties

Label 4-20e4-1.1-c2e2-0-5
Degree $4$
Conductor $160000$
Sign $1$
Analytic cond. $118.792$
Root an. cond. $3.30139$
Motivic weight $2$
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  + 15·9-s − 32·13-s + 6·17-s + 24·29-s − 100·37-s − 126·41-s + 86·49-s − 36·53-s + 52·61-s + 34·73-s + 144·81-s + 198·89-s + 268·97-s + 300·101-s − 148·109-s + 402·113-s − 480·117-s + 95·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 90·153-s + 157-s + 163-s + ⋯
L(s)  = 1  + 5/3·9-s − 2.46·13-s + 6/17·17-s + 0.827·29-s − 2.70·37-s − 3.07·41-s + 1.75·49-s − 0.679·53-s + 0.852·61-s + 0.465·73-s + 16/9·81-s + 2.22·89-s + 2.76·97-s + 2.97·101-s − 1.35·109-s + 3.55·113-s − 4.10·117-s + 0.785·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.588·153-s + 0.00636·157-s + 0.00613·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(160000\)    =    \(2^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(118.792\)
Root analytic conductor: \(3.30139\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 160000,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.855461403\)
\(L(\frac12)\) \(\approx\) \(1.855461403\)
\(L(2)\) 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 \( 1 \)
good3$C_2^2$ \( 1 - 5 p T^{2} + p^{4} T^{4} \)
7$C_2^2$ \( 1 - 86 T^{2} + p^{4} T^{4} \)
11$C_2^2$ \( 1 - 95 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 + 16 T + p^{2} T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p^{2} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 215 T^{2} + p^{4} T^{4} \)
23$C_2^2$ \( 1 + 970 T^{2} + p^{4} T^{4} \)
29$C_2$ \( ( 1 - 12 T + p^{2} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 950 T^{2} + p^{4} T^{4} \)
37$C_2$ \( ( 1 + 50 T + p^{2} T^{2} )^{2} \)
41$C_2$ \( ( 1 + 63 T + p^{2} T^{2} )^{2} \)
43$C_2^2$ \( 1 + 190 T^{2} + p^{4} T^{4} \)
47$C_2^2$ \( 1 - 4226 T^{2} + p^{4} T^{4} \)
53$C_2$ \( ( 1 + 18 T + p^{2} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 3890 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 - 26 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 7895 T^{2} + p^{4} T^{4} \)
71$C_2^2$ \( 1 - 9314 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 - 17 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 4982 T^{2} + p^{4} T^{4} \)
83$C_2^2$ \( 1 - 4031 T^{2} + p^{4} T^{4} \)
89$C_2$ \( ( 1 - 99 T + p^{2} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 134 T + p^{2} T^{2} )^{2} \)
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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

−11.40198635865589365805544405142, −10.50372702472214618828553020690, −10.14921798578907423468680900442, −10.14236960262050464043852510770, −9.727924012762431969054764348262, −8.849227871343481957829354082775, −8.769760774812327952125427820120, −7.87916282342609564133609559158, −7.45936499312498179888951972342, −7.03140744570293138424382343170, −6.87616901708233331930535329959, −6.13683405223519419191494588980, −5.21647490578412726100104650095, −4.84681120521898977794131323163, −4.69567687399046367716028770499, −3.65690840306330561242304839703, −3.28637357230908476112014918349, −2.15743717864010414133645442047, −1.83779700904590761284750709928, −0.58833791390909552448271414090, 0.58833791390909552448271414090, 1.83779700904590761284750709928, 2.15743717864010414133645442047, 3.28637357230908476112014918349, 3.65690840306330561242304839703, 4.69567687399046367716028770499, 4.84681120521898977794131323163, 5.21647490578412726100104650095, 6.13683405223519419191494588980, 6.87616901708233331930535329959, 7.03140744570293138424382343170, 7.45936499312498179888951972342, 7.87916282342609564133609559158, 8.769760774812327952125427820120, 8.849227871343481957829354082775, 9.727924012762431969054764348262, 10.14236960262050464043852510770, 10.14921798578907423468680900442, 10.50372702472214618828553020690, 11.40198635865589365805544405142

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