Properties

Label 4-21e4-1.1-c5e2-0-9
Degree $4$
Conductor $194481$
Sign $1$
Analytic cond. $5002.62$
Root an. cond. $8.41006$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 5·4-s − 33·5-s + 99·8-s − 99·10-s + 1.13e3·11-s − 925·13-s − 459·16-s − 324·17-s − 2.31e3·19-s − 165·20-s + 3.41e3·22-s − 1.59e3·23-s − 2.38e3·25-s − 2.77e3·26-s + 2.21e3·29-s − 4.29e3·31-s − 4.36e3·32-s − 972·34-s − 1.91e4·37-s − 6.93e3·38-s − 3.26e3·40-s − 1.28e4·41-s − 2.77e3·43-s + 5.68e3·44-s − 4.78e3·46-s − 2.31e4·47-s + ⋯
L(s)  = 1  + 0.530·2-s + 5/32·4-s − 0.590·5-s + 0.546·8-s − 0.313·10-s + 2.83·11-s − 1.51·13-s − 0.448·16-s − 0.271·17-s − 1.46·19-s − 0.0922·20-s + 1.50·22-s − 0.629·23-s − 0.762·25-s − 0.805·26-s + 0.489·29-s − 0.802·31-s − 0.753·32-s − 0.144·34-s − 2.29·37-s − 0.778·38-s − 0.322·40-s − 1.19·41-s − 0.228·43-s + 0.442·44-s − 0.333·46-s − 1.52·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(194481\)    =    \(3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(5002.62\)
Root analytic conductor: \(8.41006\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 194481,\ (\ :5/2, 5/2),\ 1)\)

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 \( 1 \)
7 \( 1 \)
good2$D_{4}$ \( 1 - 3 T + p^{2} T^{2} - 3 p^{5} T^{3} + p^{10} T^{4} \)
5$D_{4}$ \( 1 + 33 T + 3472 T^{2} + 33 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 - 1137 T + 645232 T^{2} - 1137 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 + 925 T + 951450 T^{2} + 925 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 + 324 T + 1502434 T^{2} + 324 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 + 2311 T + 6241998 T^{2} + 2311 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 + 1596 T + 7604206 T^{2} + 1596 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 - 2217 T + 42125014 T^{2} - 2217 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 + 4294 T + 25151367 T^{2} + 4294 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 + 19109 T + 184470078 T^{2} + 19109 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 + 12858 T + 228491122 T^{2} + 12858 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 + 2771 T + 36114396 T^{2} + 2771 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 + 23160 T + 570076618 T^{2} + 23160 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 31653 T + 896010526 T^{2} - 31653 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 - 41097 T + 896994610 T^{2} - 41097 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 42052 T + 1728407262 T^{2} + 42052 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 - 30763 T + 1698033324 T^{2} - 30763 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 + 102096 T + 6091648810 T^{2} + 102096 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 - 28577 T + 2636499108 T^{2} - 28577 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 + 18464 T + 3332046261 T^{2} + 18464 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 - 61179 T + 8589312784 T^{2} - 61179 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 - 29322 T + 11382011794 T^{2} - 29322 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 - 9791 T + 17134261944 T^{2} - 9791 p^{5} T^{3} + p^{10} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.13199842576144227096604135010, −9.712000892620380072860334888702, −9.012427557239995462750680318216, −8.823384873458009625849919547148, −8.399402264299868026734800389322, −7.66642919787712372484650821597, −7.07295666054133509186864911561, −6.87315311111943928906750493039, −6.47697299208237884606306989848, −5.92027529356180219567306530294, −5.03968112962672072890145470059, −4.75846786485297613865236781333, −3.99043836664774357386062073019, −3.95572636384637669361891774875, −3.36316645746080433214420206484, −2.24669176330310343385865794890, −1.87127008400953892077469303103, −1.31064231601552648464275750368, 0, 0, 1.31064231601552648464275750368, 1.87127008400953892077469303103, 2.24669176330310343385865794890, 3.36316645746080433214420206484, 3.95572636384637669361891774875, 3.99043836664774357386062073019, 4.75846786485297613865236781333, 5.03968112962672072890145470059, 5.92027529356180219567306530294, 6.47697299208237884606306989848, 6.87315311111943928906750493039, 7.07295666054133509186864911561, 7.66642919787712372484650821597, 8.399402264299868026734800389322, 8.823384873458009625849919547148, 9.012427557239995462750680318216, 9.712000892620380072860334888702, 10.13199842576144227096604135010

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