Properties

Label 4-24e4-1.1-c2e2-0-7
Degree $4$
Conductor $331776$
Sign $1$
Analytic cond. $246.328$
Root an. cond. $3.96167$
Motivic weight $2$
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  + 8·7-s − 16·13-s − 32·19-s + 32·25-s − 88·31-s + 68·37-s − 80·43-s − 50·49-s − 100·61-s + 16·67-s − 32·73-s + 152·79-s − 128·91-s + 352·97-s + 56·103-s − 46·121-s + 127-s + 131-s − 256·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 146·169-s + ⋯
L(s)  = 1  + 8/7·7-s − 1.23·13-s − 1.68·19-s + 1.27·25-s − 2.83·31-s + 1.83·37-s − 1.86·43-s − 1.02·49-s − 1.63·61-s + 0.238·67-s − 0.438·73-s + 1.92·79-s − 1.40·91-s + 3.62·97-s + 0.543·103-s − 0.380·121-s + 0.00787·127-s + 0.00763·131-s − 1.92·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.863·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(331776\)    =    \(2^{12} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(246.328\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{576} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 331776,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.650347501\)
\(L(\frac12)\) \(\approx\) \(1.650347501\)
\(L(2)\) 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 \( 1 \)
good5$C_2^2$ \( 1 - 32 T^{2} + p^{4} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p^{2} T^{2} )^{2} \)
11$C_2$ \( ( 1 - 14 T + p^{2} T^{2} )( 1 + 14 T + p^{2} T^{2} ) \)
13$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 416 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 + 16 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 770 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 1664 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 + 44 T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 34 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 1184 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 40 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 2782 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 4160 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 5810 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 50 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 7490 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 + 16 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 76 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 334 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 15680 T^{2} + p^{4} T^{4} \)
97$C_2$ \( ( 1 - 176 T + p^{2} T^{2} )^{2} \)
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

−10.87548642493860639251882709288, −10.38657295919216017914004676548, −9.904309879826644061867577309318, −9.303229594422882131084680659752, −9.017074122597167586942537555400, −8.585294933962246197013940123430, −7.920249674073304813048618304025, −7.75767836017893870097385507627, −7.27921972016935093144738772096, −6.48972727011012459093414816766, −6.46502233235709708214728186314, −5.50745688625015102449690600891, −5.13258638695520631463292853335, −4.63450192156492300358272123092, −4.32307144306747344205210136404, −3.48735849955437090272040029836, −2.87543926117971228935150249411, −1.93885643398718639691324804647, −1.80442980206657036402054405532, −0.47264380370924817835984330791, 0.47264380370924817835984330791, 1.80442980206657036402054405532, 1.93885643398718639691324804647, 2.87543926117971228935150249411, 3.48735849955437090272040029836, 4.32307144306747344205210136404, 4.63450192156492300358272123092, 5.13258638695520631463292853335, 5.50745688625015102449690600891, 6.46502233235709708214728186314, 6.48972727011012459093414816766, 7.27921972016935093144738772096, 7.75767836017893870097385507627, 7.920249674073304813048618304025, 8.585294933962246197013940123430, 9.017074122597167586942537555400, 9.303229594422882131084680659752, 9.904309879826644061867577309318, 10.38657295919216017914004676548, 10.87548642493860639251882709288

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