Properties

Label 4-21e4-1.1-c1e2-0-15
Degree $4$
Conductor $194481$
Sign $1$
Analytic cond. $12.4002$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s + 2·5-s + 4·8-s + 4·10-s − 2·11-s − 2·13-s + 8·16-s + 19-s + 4·20-s − 4·22-s + 5·25-s − 4·26-s − 8·29-s + 9·31-s + 8·32-s − 3·37-s + 2·38-s + 8·40-s − 20·41-s + 10·43-s − 4·44-s + 6·47-s + 10·50-s − 4·52-s + 12·53-s − 4·55-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 0.894·5-s + 1.41·8-s + 1.26·10-s − 0.603·11-s − 0.554·13-s + 2·16-s + 0.229·19-s + 0.894·20-s − 0.852·22-s + 25-s − 0.784·26-s − 1.48·29-s + 1.61·31-s + 1.41·32-s − 0.493·37-s + 0.324·38-s + 1.26·40-s − 3.12·41-s + 1.52·43-s − 0.603·44-s + 0.875·47-s + 1.41·50-s − 0.554·52-s + 1.64·53-s − 0.539·55-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(194481\)    =    \(3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(12.4002\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 194481,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.554099979\)
\(L(\frac12)\) \(\approx\) \(4.554099979\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 3 T - 64 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 16 T + 167 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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

−11.47427252918429605360581773829, −10.88381478687683212611247892420, −10.42037856594733381649778852369, −10.19108765328872676940965825392, −9.722738693879243613827129138377, −9.233153361206338198543210490442, −8.345616257938071359097484650533, −8.300831912792073410911994983002, −7.43218123963546323519361995834, −7.01842397817773169064687649106, −6.72508313428073797690881419585, −5.87295572801602683615539219319, −5.52760332090179227702685584001, −4.96952953562237227176871892073, −4.90003992433324134215765779852, −3.81829038348525041566569393420, −3.70354457972616565365145955591, −2.56147766452243104873584991151, −2.27689263339679432504867999504, −1.23016460075341789433949643971, 1.23016460075341789433949643971, 2.27689263339679432504867999504, 2.56147766452243104873584991151, 3.70354457972616565365145955591, 3.81829038348525041566569393420, 4.90003992433324134215765779852, 4.96952953562237227176871892073, 5.52760332090179227702685584001, 5.87295572801602683615539219319, 6.72508313428073797690881419585, 7.01842397817773169064687649106, 7.43218123963546323519361995834, 8.300831912792073410911994983002, 8.345616257938071359097484650533, 9.233153361206338198543210490442, 9.722738693879243613827129138377, 10.19108765328872676940965825392, 10.42037856594733381649778852369, 10.88381478687683212611247892420, 11.47427252918429605360581773829

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