Properties

Label 4-6e8-1.1-c1e2-0-4
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 3·7-s + 5·11-s − 4·13-s − 16·17-s − 4·19-s + 2·23-s + 5·25-s − 6·29-s − 7·31-s + 3·35-s − 12·37-s + 6·41-s − 2·43-s + 6·47-s + 7·49-s + 10·53-s + 5·55-s − 4·59-s + 8·61-s − 4·65-s − 10·67-s + 16·71-s + 2·73-s + 15·77-s + 16·79-s − 11·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.13·7-s + 1.50·11-s − 1.10·13-s − 3.88·17-s − 0.917·19-s + 0.417·23-s + 25-s − 1.11·29-s − 1.25·31-s + 0.507·35-s − 1.97·37-s + 0.937·41-s − 0.304·43-s + 0.875·47-s + 49-s + 1.37·53-s + 0.674·55-s − 0.520·59-s + 1.02·61-s − 0.496·65-s − 1.22·67-s + 1.89·71-s + 0.234·73-s + 1.70·77-s + 1.80·79-s − 1.20·83-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: \(0\)
Selberg data: \((4,\ 1679616,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.930008421\)
\(L(\frac12)\) \(\approx\) \(1.930008421\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 11 T + 38 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.620884626245311715179437901058, −9.349616137934282788151949832386, −8.947918434215541808881819641105, −8.665264483932739708543942344117, −8.631773009216457655289104148310, −7.80505830461215580922561570875, −7.19065266565793287183054619160, −6.90338699782206882538814898869, −6.77784400855752444409487585816, −6.25634209951146386031091629133, −5.60418252433387890042564675962, −5.17144354159188181379688716291, −4.70485412316276148733355642903, −4.20232803554501897870850557385, −4.13802191171115778905610912905, −3.30706477490523308440198843026, −2.31488220422973967779953729488, −2.05622063501945365174930143288, −1.81070112000778147368059967004, −0.55381338827197526572741634589, 0.55381338827197526572741634589, 1.81070112000778147368059967004, 2.05622063501945365174930143288, 2.31488220422973967779953729488, 3.30706477490523308440198843026, 4.13802191171115778905610912905, 4.20232803554501897870850557385, 4.70485412316276148733355642903, 5.17144354159188181379688716291, 5.60418252433387890042564675962, 6.25634209951146386031091629133, 6.77784400855752444409487585816, 6.90338699782206882538814898869, 7.19065266565793287183054619160, 7.80505830461215580922561570875, 8.631773009216457655289104148310, 8.665264483932739708543942344117, 8.947918434215541808881819641105, 9.349616137934282788151949832386, 9.620884626245311715179437901058

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