Properties

Label 4-192e2-1.1-c1e2-0-1
Degree $4$
Conductor $36864$
Sign $1$
Analytic cond. $2.35048$
Root an. cond. $1.23819$
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  − 3·9-s + 4·13-s + 10·25-s + 20·37-s + 2·49-s − 28·61-s + 20·73-s + 9·81-s − 28·97-s + 4·109-s − 12·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 9-s + 1.10·13-s + 2·25-s + 3.28·37-s + 2/7·49-s − 3.58·61-s + 2.34·73-s + 81-s − 2.84·97-s + 0.383·109-s − 1.10·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(36864\)    =    \(2^{12} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(2.35048\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{192} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 36864,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.277028929\)
\(L(\frac12)\) \(\approx\) \(1.277028929\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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

−12.62319681634606304500547076579, −12.48156897690249202858230097469, −11.68598738069570225342057783170, −11.25165812233945499027023204816, −10.84893670739889342900815061984, −10.60278849114699696559566293376, −9.708184890174179423079431268622, −9.161734831488399239895785093245, −8.937434081000418110369817580115, −8.041862585418433570099595947456, −8.031716069625640784140838134814, −7.11832588898137780415786460288, −6.30023882575175586006277492091, −6.18228802374312921582724731063, −5.38158919701726417869764232295, −4.70291808743084588513594320992, −4.02421232543024795597776629105, −3.10170155764302832632860294951, −2.60182485234272276047494052157, −1.13039910110545379294649008287, 1.13039910110545379294649008287, 2.60182485234272276047494052157, 3.10170155764302832632860294951, 4.02421232543024795597776629105, 4.70291808743084588513594320992, 5.38158919701726417869764232295, 6.18228802374312921582724731063, 6.30023882575175586006277492091, 7.11832588898137780415786460288, 8.031716069625640784140838134814, 8.041862585418433570099595947456, 8.937434081000418110369817580115, 9.161734831488399239895785093245, 9.708184890174179423079431268622, 10.60278849114699696559566293376, 10.84893670739889342900815061984, 11.25165812233945499027023204816, 11.68598738069570225342057783170, 12.48156897690249202858230097469, 12.62319681634606304500547076579

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