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  + 12·3-s + 50·5-s − 52·7-s + 138·9-s − 560·11-s + 1.38e3·13-s + 600·15-s + 148·17-s + 1.00e3·19-s − 624·21-s + 2.45e3·23-s + 1.87e3·25-s + 4.50e3·27-s + 1.34e3·29-s + 2.24e3·31-s − 6.72e3·33-s − 2.60e3·35-s − 5.94e3·37-s + 1.66e4·39-s + 2.30e4·41-s − 1.76e4·43-s + 6.90e3·45-s + 2.90e3·47-s − 2.69e4·49-s + 1.77e3·51-s − 5.41e3·53-s − 2.80e4·55-s + ⋯
L(s)  = 1  + 0.769·3-s + 0.894·5-s − 0.401·7-s + 0.567·9-s − 1.39·11-s + 2.27·13-s + 0.688·15-s + 0.124·17-s + 0.635·19-s − 0.308·21-s + 0.966·23-s + 3/5·25-s + 1.18·27-s + 0.295·29-s + 0.420·31-s − 1.07·33-s − 0.358·35-s − 0.713·37-s + 1.75·39-s + 2.14·41-s − 1.45·43-s + 0.507·45-s + 0.192·47-s − 1.60·49-s + 0.0956·51-s − 0.264·53-s − 1.24·55-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

\(L(3)\) \(\approx\) \(4.20633\)
\(L(\frac12)\) \(\approx\) \(4.20633\)
\(L(\frac{7}{2})\) 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$C_1$ \( ( 1 - p^{2} T )^{2} \)
good3$D_{4}$ \( 1 - 4 p T + 2 p T^{2} - 4 p^{6} T^{3} + p^{10} T^{4} \)
7$D_{4}$ \( 1 + 52 T + 29646 T^{2} + 52 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 + 560 T + 381926 T^{2} + 560 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 - 1388 T + 1149918 T^{2} - 1388 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 - 148 T - 795706 T^{2} - 148 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 - 1000 T + 4533462 T^{2} - 1000 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 - 2452 T + 6569198 T^{2} - 2452 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 - 1340 T - 8758306 T^{2} - 1340 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 - 2248 T + 57017022 T^{2} - 2248 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 + 5940 T + 123434318 T^{2} + 5940 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 - 23076 T + 352280470 T^{2} - 23076 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 + 17684 T + 312898614 T^{2} + 17684 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 - 2908 T + 56660030 T^{2} - 2908 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 + 5412 T + 693247822 T^{2} + 5412 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 + 62584 T + 2277965606 T^{2} + 62584 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 - 14108 T + 1110042462 T^{2} - 14108 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 - 85412 T + 4371910566 T^{2} - 85412 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 + 47208 T + 4011779662 T^{2} + 47208 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 + 924 p T + 4400780438 T^{2} + 924 p^{6} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 - 65904 T + 3994274078 T^{2} - 65904 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 + 108724 T + 10572459494 T^{2} + 108724 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 + 55020 T + 10818978262 T^{2} + 55020 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 - 147668 T + 11612429670 T^{2} - 147668 p^{5} T^{3} + p^{10} T^{4} \)
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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

−13.52264286000985000523930192144, −13.22141430387596731804573087869, −12.92523797441355372694644036642, −12.30307350352262584450903273939, −11.19926942396042857658251164258, −10.95955500367018888610578644902, −10.24098021989322024302160726647, −9.843976300791482150622432771857, −9.100889102275110912313548705786, −8.622647609257301174494213539937, −8.119619806704461866264543309587, −7.38097935374464368211514605799, −6.54488774356521669943509936710, −6.03448109439684576404105985846, −5.27784775488046631804627170362, −4.47386861212425605095010252040, −3.22327427245931833527234206242, −3.00116954620236432032596373564, −1.76865176153048056221471300631, −0.894891540791953808883245827389, 0.894891540791953808883245827389, 1.76865176153048056221471300631, 3.00116954620236432032596373564, 3.22327427245931833527234206242, 4.47386861212425605095010252040, 5.27784775488046631804627170362, 6.03448109439684576404105985846, 6.54488774356521669943509936710, 7.38097935374464368211514605799, 8.119619806704461866264543309587, 8.622647609257301174494213539937, 9.100889102275110912313548705786, 9.843976300791482150622432771857, 10.24098021989322024302160726647, 10.95955500367018888610578644902, 11.19926942396042857658251164258, 12.30307350352262584450903273939, 12.92523797441355372694644036642, 13.22141430387596731804573087869, 13.52264286000985000523930192144

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