Properties

Label 4-6e8-1.1-c1e2-0-32
Degree $4$
Conductor $1679616$
Sign $1$
Analytic cond. $107.093$
Root an. cond. $3.21692$
Motivic weight $1$
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  − 4·5-s − 3·7-s + 4·11-s − 13-s − 8·17-s + 2·19-s + 4·23-s + 5·25-s − 4·31-s + 12·35-s − 18·37-s − 8·43-s − 12·47-s + 7·49-s − 16·53-s − 16·55-s + 4·59-s + 5·61-s + 4·65-s + 11·67-s − 16·71-s + 2·73-s − 12·77-s − 5·79-s + 8·83-s + 32·85-s + 24·89-s + ⋯
L(s)  = 1  − 1.78·5-s − 1.13·7-s + 1.20·11-s − 0.277·13-s − 1.94·17-s + 0.458·19-s + 0.834·23-s + 25-s − 0.718·31-s + 2.02·35-s − 2.95·37-s − 1.21·43-s − 1.75·47-s + 49-s − 2.19·53-s − 2.15·55-s + 0.520·59-s + 0.640·61-s + 0.496·65-s + 1.34·67-s − 1.89·71-s + 0.234·73-s − 1.36·77-s − 0.562·79-s + 0.878·83-s + 3.47·85-s + 2.54·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1679616\)    =    \(2^{8} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(107.093\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1296} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1679616,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 8 T - 19 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \)
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

−9.500649251207420568219218988039, −9.099326134506888976388389403865, −8.485723398831119828084220288601, −8.445952931243158883147403713774, −7.79823939270221944414961945552, −7.36656014966167808402898350612, −6.80814427043802261992235946776, −6.63157983898903221583972715568, −6.58988464463012920252897554945, −5.67241571895452984371100486874, −4.94500255728065287148824078363, −4.87336213796623534854101082262, −3.95468530329711412549310601519, −3.90451297928513085658154998130, −3.34050857360392175056115680779, −3.02658042727998162700121981244, −2.07877301648297611464705464658, −1.40099024325289169355249570498, 0, 0, 1.40099024325289169355249570498, 2.07877301648297611464705464658, 3.02658042727998162700121981244, 3.34050857360392175056115680779, 3.90451297928513085658154998130, 3.95468530329711412549310601519, 4.87336213796623534854101082262, 4.94500255728065287148824078363, 5.67241571895452984371100486874, 6.58988464463012920252897554945, 6.63157983898903221583972715568, 6.80814427043802261992235946776, 7.36656014966167808402898350612, 7.79823939270221944414961945552, 8.445952931243158883147403713774, 8.485723398831119828084220288601, 9.099326134506888976388389403865, 9.500649251207420568219218988039

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