Properties

Degree $4$
Conductor $6400$
Sign $1$
Motivic weight $5$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 110·5-s + 290·9-s + 296·11-s + 4.44e3·19-s + 8.97e3·25-s + 540·29-s + 4.09e3·31-s − 4.79e3·41-s + 3.19e4·45-s + 8.65e3·49-s + 3.25e4·55-s − 7.94e4·59-s − 8.45e4·61-s + 8.49e3·71-s + 7.05e4·79-s + 2.50e4·81-s + 1.70e5·89-s + 4.88e5·95-s + 8.58e4·99-s − 8.59e3·101-s + 7.19e4·109-s − 2.56e5·121-s + 6.43e5·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1.96·5-s + 1.19·9-s + 0.737·11-s + 2.82·19-s + 2.87·25-s + 0.119·29-s + 0.765·31-s − 0.445·41-s + 2.34·45-s + 0.514·49-s + 1.45·55-s − 2.97·59-s − 2.91·61-s + 0.200·71-s + 1.27·79-s + 0.424·81-s + 2.28·89-s + 5.55·95-s + 0.880·99-s − 0.0838·101-s + 0.580·109-s − 1.59·121-s + 3.68·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6400\)    =    \(2^{8} \cdot 5^{2}\)
Sign: $1$
Motivic weight: \(5\)
Character: induced by $\chi_{80} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6400,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(5.09622\)
\(L(\frac12)\) \(\approx\) \(5.09622\)
\(L(\frac{7}{2})\) 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$C_2$ \( 1 - 22 p T + p^{5} T^{2} \)
good3$C_2^2$ \( 1 - 290 T^{2} + p^{10} T^{4} \)
7$C_2^2$ \( 1 - 8650 T^{2} + p^{10} T^{4} \)
11$C_2$ \( ( 1 - 148 T + p^{5} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 274730 T^{2} + p^{10} T^{4} \)
17$C_2^2$ \( 1 + 1354590 T^{2} + p^{10} T^{4} \)
19$C_2$ \( ( 1 - 2220 T + p^{5} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 11320170 T^{2} + p^{10} T^{4} \)
29$C_2$ \( ( 1 - 270 T + p^{5} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2048 T + p^{5} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 119573530 T^{2} + p^{10} T^{4} \)
41$C_2$ \( ( 1 + 2398 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 288754450 T^{2} + p^{10} T^{4} \)
47$C_2^2$ \( 1 - 344584890 T^{2} + p^{10} T^{4} \)
53$C_2^2$ \( 1 - 827605690 T^{2} + p^{10} T^{4} \)
59$C_2$ \( ( 1 + 39740 T + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 42298 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 1669968610 T^{2} + p^{10} T^{4} \)
71$C_2$ \( ( 1 - 4248 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 3239892370 T^{2} + p^{10} T^{4} \)
79$C_2$ \( ( 1 - 35280 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 7103795010 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 - 85210 T + p^{5} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 7720618690 T^{2} + p^{10} T^{4} \)
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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

−13.58248784421289701729257868103, −13.50659965957190715628203683561, −12.56767908793262022207124289458, −12.19671509682654663964408871948, −11.59846747118026925394831266567, −10.67475989277613844932529892691, −10.27991311618157684912840832220, −9.723601451032213785933466091870, −9.246543034284442442540938365465, −9.051773555324886597155218867365, −7.70784665236488771735719467181, −7.34926583302124053202932681555, −6.37389826249377757064204605115, −6.16985539287195084585772662105, −5.12958350681591005540140504081, −4.79017797052126808819223439258, −3.50316828688008904089271131000, −2.66908325633065254032997145101, −1.43484698478187625651289244074, −1.22449497840952489148991991128, 1.22449497840952489148991991128, 1.43484698478187625651289244074, 2.66908325633065254032997145101, 3.50316828688008904089271131000, 4.79017797052126808819223439258, 5.12958350681591005540140504081, 6.16985539287195084585772662105, 6.37389826249377757064204605115, 7.34926583302124053202932681555, 7.70784665236488771735719467181, 9.051773555324886597155218867365, 9.246543034284442442540938365465, 9.723601451032213785933466091870, 10.27991311618157684912840832220, 10.67475989277613844932529892691, 11.59846747118026925394831266567, 12.19671509682654663964408871948, 12.56767908793262022207124289458, 13.50659965957190715628203683561, 13.58248784421289701729257868103

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