Properties

Label 4-147e2-1.1-c3e2-0-1
Degree $4$
Conductor $21609$
Sign $1$
Analytic cond. $75.2257$
Root an. cond. $2.94504$
Motivic weight $3$
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  − 9·3-s − 8·4-s + 54·9-s + 72·12-s + 270·19-s + 125·25-s − 243·27-s − 270·31-s − 432·36-s + 110·37-s + 1.04e3·43-s − 2.43e3·57-s − 1.62e3·61-s + 512·64-s + 880·67-s − 648·73-s − 1.12e3·75-s − 2.16e3·76-s − 884·79-s + 729·81-s + 2.43e3·93-s − 1.00e3·100-s + 1.78e3·103-s + 1.94e3·108-s − 646·109-s − 990·111-s − 1.33e3·121-s + ⋯
L(s)  = 1  − 1.73·3-s − 4-s + 2·9-s + 1.73·12-s + 3.26·19-s + 25-s − 1.73·27-s − 1.56·31-s − 2·36-s + 0.488·37-s + 3.68·43-s − 5.64·57-s − 3.40·61-s + 64-s + 1.60·67-s − 1.03·73-s − 1.73·75-s − 3.26·76-s − 1.25·79-s + 81-s + 2.70·93-s − 100-s + 1.70·103-s + 1.73·108-s − 0.567·109-s − 0.846·111-s − 121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21609\)    =    \(3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(75.2257\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{147} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21609,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8333497756\)
\(L(\frac12)\) \(\approx\) \(0.8333497756\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + p^{2} T + p^{3} T^{2} \)
7 \( 1 \)
good2$C_2^2$ \( 1 + p^{3} T^{2} + p^{6} T^{4} \)
5$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + p^{3} T^{2} + p^{6} T^{4} \)
13$C_2$ \( ( 1 - 70 T + p^{3} T^{2} )( 1 + 70 T + p^{3} T^{2} ) \)
17$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 - 163 T + p^{3} T^{2} )( 1 - 107 T + p^{3} T^{2} ) \)
23$C_2^2$ \( 1 + p^{3} T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 19 T + p^{3} T^{2} )( 1 + 289 T + p^{3} T^{2} ) \)
37$C_2$ \( ( 1 - 433 T + p^{3} T^{2} )( 1 + 323 T + p^{3} T^{2} ) \)
41$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 520 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + p^{3} T^{2} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
61$C_2$ \( ( 1 + 719 T + p^{3} T^{2} )( 1 + 901 T + p^{3} T^{2} ) \)
67$C_2$ \( ( 1 - 1007 T + p^{3} T^{2} )( 1 + 127 T + p^{3} T^{2} ) \)
71$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 271 T + p^{3} T^{2} )( 1 + 919 T + p^{3} T^{2} ) \)
79$C_2$ \( ( 1 - 503 T + p^{3} T^{2} )( 1 + 1387 T + p^{3} T^{2} ) \)
83$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
97$C_2$ \( ( 1 - 1330 T + p^{3} T^{2} )( 1 + 1330 T + p^{3} T^{2} ) \)
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

−12.63873196349979223192622094904, −12.46510193891465516402282067257, −11.59891360461091762294151678547, −11.56650731253101578977590132992, −10.76051627745620709451713289877, −10.54790947751493021873182081754, −9.669719160580259528917779851105, −9.274441078007337872686550887648, −9.098921141686755449508402713263, −7.934377079934207405420513607946, −7.23527516182156214867234965693, −7.20485465144881848729034944209, −5.97264956388113791407930017405, −5.73414961016909173115037478452, −5.08864969674775856924329194285, −4.64221855032999634865341569835, −3.91029483423987447085577614417, −2.94844048321400508670640805001, −1.27087257537561382827861373003, −0.59114164568288093396210999289, 0.59114164568288093396210999289, 1.27087257537561382827861373003, 2.94844048321400508670640805001, 3.91029483423987447085577614417, 4.64221855032999634865341569835, 5.08864969674775856924329194285, 5.73414961016909173115037478452, 5.97264956388113791407930017405, 7.20485465144881848729034944209, 7.23527516182156214867234965693, 7.934377079934207405420513607946, 9.098921141686755449508402713263, 9.274441078007337872686550887648, 9.669719160580259528917779851105, 10.54790947751493021873182081754, 10.76051627745620709451713289877, 11.56650731253101578977590132992, 11.59891360461091762294151678547, 12.46510193891465516402282067257, 12.63873196349979223192622094904

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