Properties

Degree $4$
Conductor $160000$
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  + 20·3-s + 200·7-s + 55·9-s + 196·11-s + 360·13-s − 1.49e3·17-s + 3.18e3·19-s + 4.00e3·21-s + 1.56e3·23-s − 940·27-s − 3.92e3·29-s + 1.09e3·31-s + 3.92e3·33-s − 2.02e3·37-s + 7.20e3·39-s + 2.77e4·41-s − 3.00e3·43-s + 2.57e4·47-s − 2.65e3·49-s − 2.98e4·51-s + 2.69e4·53-s + 6.36e4·57-s − 1.19e4·59-s − 2.43e4·61-s + 1.10e4·63-s − 4.00e4·67-s + 3.12e4·69-s + ⋯
L(s)  = 1  + 1.28·3-s + 1.54·7-s + 0.226·9-s + 0.488·11-s + 0.590·13-s − 1.25·17-s + 2.02·19-s + 1.97·21-s + 0.614·23-s − 0.248·27-s − 0.865·29-s + 0.204·31-s + 0.626·33-s − 0.242·37-s + 0.758·39-s + 2.57·41-s − 0.247·43-s + 1.70·47-s − 0.157·49-s − 1.60·51-s + 1.31·53-s + 2.59·57-s − 0.447·59-s − 0.839·61-s + 0.349·63-s − 1.09·67-s + 0.788·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(160000\)    =    \(2^{8} \cdot 5^{4}\)
Sign: $1$
Motivic weight: \(5\)
Character: induced by $\chi_{400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 160000,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(8.728292643\)
\(L(\frac12)\) \(\approx\) \(8.728292643\)
\(L(\frac{7}{2})\) 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 \( 1 \)
good3$D_{4}$ \( 1 - 20 T + 115 p T^{2} - 20 p^{5} T^{3} + p^{10} T^{4} \)
7$D_{4}$ \( 1 - 200 T + 42650 T^{2} - 200 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 - 196 T + 181081 T^{2} - 196 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 - 360 T + 713290 T^{2} - 360 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 + 1490 T + 2280355 T^{2} + 1490 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 - 3180 T + 7185073 T^{2} - 3180 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 - 1560 T + 13472410 T^{2} - 1560 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 + 3920 T + 44478298 T^{2} + 3920 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 - 1096 T - 15343894 T^{2} - 1096 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 + 2020 T + 130823790 T^{2} + 2020 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 - 27754 T + 414643531 T^{2} - 27754 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 + 3000 T + 23431750 T^{2} + 3000 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 - 25760 T + 480952270 T^{2} - 25760 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 26980 T + 984299470 T^{2} - 26980 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 + 11960 T + 1234152598 T^{2} + 11960 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 24396 T + 1596983806 T^{2} + 24396 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 + 40060 T + 2949898505 T^{2} + 40060 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 - 87296 T + 5453356606 T^{2} - 87296 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 - 70290 T + 5306812435 T^{2} - 70290 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 + 65480 T + 5707696298 T^{2} + 65480 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 - 92580 T + 9976491505 T^{2} - 92580 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 + 72810 T + 5926578523 T^{2} + 72810 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 - 126140 T + 13936294470 T^{2} - 126140 p^{5} T^{3} + p^{10} T^{4} \)
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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

−10.89076753273590244511245547042, −10.21761518961975166188871211990, −9.427370150805717267392975804051, −9.238665763092530542728124660557, −8.867865931595605439979241255843, −8.475789529566974499214828139504, −7.86731576539804107883479821291, −7.65835145371160270939437195432, −7.17305070309182479320245476207, −6.54663478839393067463603638661, −5.75541962469641083996100860404, −5.46096694243240174390212588375, −4.60557277768985972496482298225, −4.41076396743122626779521326921, −3.52202524990560935427513190270, −3.20797351358336802516694669401, −2.33922417839937057309173227973, −2.04214349764026624114689933067, −1.15923660263639035114368770645, −0.74531022172690809737111640214, 0.74531022172690809737111640214, 1.15923660263639035114368770645, 2.04214349764026624114689933067, 2.33922417839937057309173227973, 3.20797351358336802516694669401, 3.52202524990560935427513190270, 4.41076396743122626779521326921, 4.60557277768985972496482298225, 5.46096694243240174390212588375, 5.75541962469641083996100860404, 6.54663478839393067463603638661, 7.17305070309182479320245476207, 7.65835145371160270939437195432, 7.86731576539804107883479821291, 8.475789529566974499214828139504, 8.867865931595605439979241255843, 9.238665763092530542728124660557, 9.427370150805717267392975804051, 10.21761518961975166188871211990, 10.89076753273590244511245547042

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