Properties

Label 4-15e4-1.1-c5e2-0-15
Degree $4$
Conductor $50625$
Sign $1$
Analytic cond. $1302.22$
Root an. cond. $6.00719$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s + 15·4-s + 200·7-s − 185·8-s + 196·11-s − 360·13-s − 1.00e3·14-s + 551·16-s − 1.49e3·17-s − 3.18e3·19-s − 980·22-s − 1.56e3·23-s + 1.80e3·26-s + 3.00e3·28-s + 3.92e3·29-s − 1.09e3·31-s + 1.81e3·32-s + 7.45e3·34-s + 2.02e3·37-s + 1.59e4·38-s − 2.77e4·41-s − 3.00e3·43-s + 2.94e3·44-s + 7.80e3·46-s − 2.57e4·47-s − 2.65e3·49-s − 5.40e3·52-s + ⋯
L(s)  = 1  − 0.883·2-s + 0.468·4-s + 1.54·7-s − 1.02·8-s + 0.488·11-s − 0.590·13-s − 1.36·14-s + 0.538·16-s − 1.25·17-s − 2.02·19-s − 0.431·22-s − 0.614·23-s + 0.522·26-s + 0.723·28-s + 0.865·29-s − 0.204·31-s + 0.313·32-s + 1.10·34-s + 0.242·37-s + 1.78·38-s − 2.57·41-s − 0.247·43-s + 0.228·44-s + 0.543·46-s − 1.70·47-s − 0.157·49-s − 0.276·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50625\)    =    \(3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(1302.22\)
Root analytic conductor: \(6.00719\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 50625,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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)$
bad3 \( 1 \)
5 \( 1 \)
good2$D_{4}$ \( 1 + 5 T + 5 p T^{2} + 5 p^{5} T^{3} + p^{10} T^{4} \)
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

−11.10390106665314711031466504522, −10.71323935629781122022195666280, −10.05933169425057618766115230660, −9.747062725858678564557160800624, −8.892714306688140661548242242063, −8.729325419245455384321399121983, −8.209856992381969896322049509054, −7.997677551596860814143396759278, −7.04515104322681489396525784513, −6.53537251888497728081225124152, −6.32500414625735130685516540660, −5.34789786502697058613036159032, −4.63103306426382798966951505982, −4.43038800804695079457307358329, −3.43692128757379197576022710081, −2.46989812978575873968196072541, −1.94758629757877113434697467608, −1.38295647189362516673025387841, 0, 0, 1.38295647189362516673025387841, 1.94758629757877113434697467608, 2.46989812978575873968196072541, 3.43692128757379197576022710081, 4.43038800804695079457307358329, 4.63103306426382798966951505982, 5.34789786502697058613036159032, 6.32500414625735130685516540660, 6.53537251888497728081225124152, 7.04515104322681489396525784513, 7.997677551596860814143396759278, 8.209856992381969896322049509054, 8.729325419245455384321399121983, 8.892714306688140661548242242063, 9.747062725858678564557160800624, 10.05933169425057618766115230660, 10.71323935629781122022195666280, 11.10390106665314711031466504522

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