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  − 76·5-s + 950·13-s − 3.05e3·17-s + 2.65e3·25-s − 950·37-s − 9.90e3·41-s − 3.29e4·53-s + 1.09e5·61-s − 7.22e4·65-s − 1.08e5·73-s − 5.90e4·81-s + 2.31e5·85-s − 2.52e5·97-s + 1.96e5·101-s − 3.62e5·113-s + 3.22e5·121-s + 3.60e4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.51e5·169-s + ⋯
L(s)  = 1  − 1.35·5-s + 1.55·13-s − 2.55·17-s + 0.848·25-s − 0.114·37-s − 0.920·41-s − 1.61·53-s + 3.78·61-s − 2.11·65-s − 2.39·73-s − 81-s + 3.47·85-s − 2.72·97-s + 1.91·101-s − 2.67·113-s + 2·121-s + 0.206·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 1.21·169-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

\(L(3)\) \(\approx\) \(0.943246\)
\(L(\frac12)\) \(\approx\) \(0.943246\)
\(L(\frac{7}{2})\) 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$C_2$ \( 1 + 76 T + p^{5} T^{2} \)
good3$C_2^2$ \( 1 + p^{10} T^{4} \)
7$C_2^2$ \( 1 + p^{10} T^{4} \)
11$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 1194 T + p^{5} T^{2} )( 1 + 244 T + p^{5} T^{2} ) \)
17$C_2$ \( ( 1 + 808 T + p^{5} T^{2} )( 1 + 2242 T + p^{5} T^{2} ) \)
19$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
23$C_2^2$ \( 1 + p^{10} T^{4} \)
29$C_2$ \( ( 1 - 2950 T + p^{5} T^{2} )( 1 + 2950 T + p^{5} T^{2} ) \)
31$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 11292 T + p^{5} T^{2} )( 1 + 12242 T + p^{5} T^{2} ) \)
41$C_2$ \( ( 1 + 4952 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 + p^{10} T^{4} \)
47$C_2^2$ \( 1 + p^{10} T^{4} \)
53$C_2$ \( ( 1 - 7294 T + p^{5} T^{2} )( 1 + 40244 T + p^{5} T^{2} ) \)
59$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 54948 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 + p^{10} T^{4} \)
71$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
73$C_2$ \( ( 1 + 20144 T + p^{5} T^{2} )( 1 + 88806 T + p^{5} T^{2} ) \)
79$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 + p^{10} T^{4} \)
89$C_2$ \( ( 1 - 51050 T + p^{5} T^{2} )( 1 + 51050 T + p^{5} T^{2} ) \)
97$C_2$ \( ( 1 + 92142 T + p^{5} T^{2} )( 1 + 160808 T + p^{5} T^{2} ) \)
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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

−13.50841718049237659912873679449, −13.12907050789952461624854822074, −12.75447727613754868588287289644, −11.81486564221855946187152761869, −11.45810039051531883711777955023, −11.09472525616520278457004079384, −10.62519063083389831877286470180, −9.780663943066431139120679917369, −8.925660761667735830555971901500, −8.496360791828485765075935561412, −8.204744654165001960618709216293, −7.20041439609260595093301767396, −6.75290159913160538177572989232, −6.11406684518511628716920339591, −5.08622599488707583938824691728, −4.19808702174283378467146342239, −3.89100766632856315079018549205, −2.87274957828654952536962508429, −1.69043278917849964065272530981, −0.41643144844483045647492667854, 0.41643144844483045647492667854, 1.69043278917849964065272530981, 2.87274957828654952536962508429, 3.89100766632856315079018549205, 4.19808702174283378467146342239, 5.08622599488707583938824691728, 6.11406684518511628716920339591, 6.75290159913160538177572989232, 7.20041439609260595093301767396, 8.204744654165001960618709216293, 8.496360791828485765075935561412, 8.925660761667735830555971901500, 9.780663943066431139120679917369, 10.62519063083389831877286470180, 11.09472525616520278457004079384, 11.45810039051531883711777955023, 11.81486564221855946187152761869, 12.75447727613754868588287289644, 13.12907050789952461624854822074, 13.50841718049237659912873679449

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