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  − 90·5-s + 90·9-s − 504·11-s + 440·19-s + 4.97e3·25-s − 1.38e4·29-s − 1.35e4·31-s − 396·41-s − 8.10e3·45-s + 3.00e4·49-s + 4.53e4·55-s + 4.93e4·59-s − 1.13e4·61-s − 1.06e5·71-s − 1.03e5·79-s − 5.09e4·81-s − 1.99e4·89-s − 3.96e4·95-s − 4.53e4·99-s − 2.18e5·101-s − 4.20e4·109-s − 1.31e5·121-s − 1.66e5·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1.60·5-s + 0.370·9-s − 1.25·11-s + 0.279·19-s + 1.59·25-s − 3.06·29-s − 2.52·31-s − 0.0367·41-s − 0.596·45-s + 1.78·49-s + 2.02·55-s + 1.84·59-s − 0.392·61-s − 2.51·71-s − 1.87·79-s − 0.862·81-s − 0.267·89-s − 0.450·95-s − 0.465·99-s − 2.12·101-s − 0.338·109-s − 0.817·121-s − 0.953·125-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

\(L(3)\) \(\approx\) \(0.344221\)
\(L(\frac12)\) \(\approx\) \(0.344221\)
\(L(\frac{7}{2})\) 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$C_2$ \( 1 + 18 p T + p^{5} T^{2} \)
good3$C_2$ \( ( 1 - 8 p T + p^{5} T^{2} )( 1 + 8 p T + p^{5} T^{2} ) \)
7$C_2^2$ \( 1 - 30050 T^{2} + p^{10} T^{4} \)
11$C_2$ \( ( 1 + 252 T + p^{5} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 728330 T^{2} + p^{10} T^{4} \)
17$C_2^2$ \( 1 - 2363810 T^{2} + p^{10} T^{4} \)
19$C_2$ \( ( 1 - 220 T + p^{5} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 6946370 T^{2} + p^{10} T^{4} \)
29$C_2$ \( ( 1 + 6930 T + p^{5} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 6752 T + p^{5} T^{2} )^{2} \)
37$C_2^2$ \( 1 + 56462470 T^{2} + p^{10} T^{4} \)
41$C_2$ \( ( 1 + 198 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 293842250 T^{2} + p^{10} T^{4} \)
47$C_2^2$ \( 1 - 347593490 T^{2} + p^{10} T^{4} \)
53$C_2^2$ \( 1 - 802472090 T^{2} + p^{10} T^{4} \)
59$C_2$ \( ( 1 - 24660 T + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 5698 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 795787610 T^{2} + p^{10} T^{4} \)
71$C_2$ \( ( 1 + 53352 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 883886830 T^{2} + p^{10} T^{4} \)
79$C_2$ \( ( 1 + 51920 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 4053674810 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 + 9990 T + p^{5} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 6923133890 T^{2} + p^{10} T^{4} \)
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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

−14.49752594411039763082254060543, −12.90983674890392720471875134896, −12.85937452426659599286633648871, −11.90777150247344623859242650377, −11.58005729088650790866170375174, −10.81171708991485132610084415289, −10.73355844752777835638069157217, −9.770690845478940023450445049360, −9.095376398425732577586925992651, −8.540671969408024928511524549821, −7.75938724834058794437686308689, −7.25338803591288837699092310252, −7.20400959014858528732836540970, −5.58315848127763899319032472826, −5.43403235251834930132038685870, −4.16887342677639859435482830729, −3.83913336687543553481393853115, −2.90215463686548024180531290082, −1.72972365910079842681397333888, −0.24545829951886585968100040249, 0.24545829951886585968100040249, 1.72972365910079842681397333888, 2.90215463686548024180531290082, 3.83913336687543553481393853115, 4.16887342677639859435482830729, 5.43403235251834930132038685870, 5.58315848127763899319032472826, 7.20400959014858528732836540970, 7.25338803591288837699092310252, 7.75938724834058794437686308689, 8.540671969408024928511524549821, 9.095376398425732577586925992651, 9.770690845478940023450445049360, 10.73355844752777835638069157217, 10.81171708991485132610084415289, 11.58005729088650790866170375174, 11.90777150247344623859242650377, 12.85937452426659599286633648871, 12.90983674890392720471875134896, 14.49752594411039763082254060543

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