Properties

Label 4-2592e2-1.1-c1e2-0-17
Degree $4$
Conductor $6718464$
Sign $1$
Analytic cond. $428.375$
Root an. cond. $4.54942$
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  + 2·5-s + 3·7-s − 6·11-s + 3·13-s + 4·17-s + 6·19-s − 6·23-s + 5·25-s − 8·29-s + 6·35-s + 14·37-s + 8·41-s − 12·43-s − 6·47-s + 7·49-s − 8·53-s − 12·55-s − 6·59-s + 61-s + 6·65-s − 3·67-s + 24·71-s − 30·73-s − 18·77-s + 9·79-s + 12·83-s + 8·85-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.13·7-s − 1.80·11-s + 0.832·13-s + 0.970·17-s + 1.37·19-s − 1.25·23-s + 25-s − 1.48·29-s + 1.01·35-s + 2.30·37-s + 1.24·41-s − 1.82·43-s − 0.875·47-s + 49-s − 1.09·53-s − 1.61·55-s − 0.781·59-s + 0.128·61-s + 0.744·65-s − 0.366·67-s + 2.84·71-s − 3.51·73-s − 2.05·77-s + 1.01·79-s + 1.31·83-s + 0.867·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6718464\)    =    \(2^{10} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(428.375\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2592} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6718464,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.460560668\)
\(L(\frac12)\) \(\approx\) \(3.460560668\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 8 T + 35 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 12 T + 101 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 9 T + 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 9 T - 16 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
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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

−9.269390723919470331617353314788, −8.574173826791273177315229619597, −8.165881471473004480257226304822, −7.88734252745482122359051764407, −7.67978462419480662080979097611, −7.44110902636368109466642316459, −6.75468025923102268424384127530, −6.18150064526574831581303777281, −5.87537734820888801624337279257, −5.64684826047703103301494899886, −5.16764215441323456528782305598, −4.86826274308701657207675479948, −4.50387781527278626018123577342, −3.78821322392808462222362090263, −3.30679812461732730469215251396, −2.90831620335266180761923725598, −2.30892897976184525098270299751, −1.84446758209790124498408074732, −1.35340753268622722761389380200, −0.63187051573737499434622558675, 0.63187051573737499434622558675, 1.35340753268622722761389380200, 1.84446758209790124498408074732, 2.30892897976184525098270299751, 2.90831620335266180761923725598, 3.30679812461732730469215251396, 3.78821322392808462222362090263, 4.50387781527278626018123577342, 4.86826274308701657207675479948, 5.16764215441323456528782305598, 5.64684826047703103301494899886, 5.87537734820888801624337279257, 6.18150064526574831581303777281, 6.75468025923102268424384127530, 7.44110902636368109466642316459, 7.67978462419480662080979097611, 7.88734252745482122359051764407, 8.165881471473004480257226304822, 8.574173826791273177315229619597, 9.269390723919470331617353314788

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