Properties

Label 4-6e8-1.1-c1e2-0-18
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  + 3·5-s − 7-s + 3·11-s + 4·13-s − 4·19-s + 6·23-s + 5·25-s + 6·29-s + 5·31-s − 3·35-s + 4·37-s − 6·41-s − 10·43-s − 6·47-s + 7·49-s − 18·53-s + 9·55-s − 12·59-s − 8·61-s + 12·65-s + 14·67-s − 14·73-s − 3·77-s + 8·79-s + 3·83-s + 36·89-s − 4·91-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.377·7-s + 0.904·11-s + 1.10·13-s − 0.917·19-s + 1.25·23-s + 25-s + 1.11·29-s + 0.898·31-s − 0.507·35-s + 0.657·37-s − 0.937·41-s − 1.52·43-s − 0.875·47-s + 49-s − 2.47·53-s + 1.21·55-s − 1.56·59-s − 1.02·61-s + 1.48·65-s + 1.71·67-s − 1.63·73-s − 0.341·77-s + 0.900·79-s + 0.329·83-s + 3.81·89-s − 0.419·91-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

\(L(1)\) \(\approx\) \(3.322398839\)
\(L(\frac12)\) \(\approx\) \(3.322398839\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 3 T - 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
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

−9.777757741237360002339327334269, −9.435276958998546136666193570814, −9.148643611844237141857628531640, −8.702492818396631087295748262906, −8.301573397976511241873904350283, −8.063439338348438485884420609619, −7.20919233048888620624394378167, −6.89098322128392487308365037656, −6.25727561079850022886066921031, −6.16009480359902246842373624528, −6.14468672509763761737082098870, −5.11651406377277662159806782209, −4.74319571739044989544841252986, −4.53568080039930419362873439029, −3.52973558185152318059097053045, −3.32954375323983113209664340505, −2.75555679847085373060118928734, −1.94029362432280859156290811229, −1.54328551351828958111255489604, −0.808117356047826066913450692885, 0.808117356047826066913450692885, 1.54328551351828958111255489604, 1.94029362432280859156290811229, 2.75555679847085373060118928734, 3.32954375323983113209664340505, 3.52973558185152318059097053045, 4.53568080039930419362873439029, 4.74319571739044989544841252986, 5.11651406377277662159806782209, 6.14468672509763761737082098870, 6.16009480359902246842373624528, 6.25727561079850022886066921031, 6.89098322128392487308365037656, 7.20919233048888620624394378167, 8.063439338348438485884420609619, 8.301573397976511241873904350283, 8.702492818396631087295748262906, 9.148643611844237141857628531640, 9.435276958998546136666193570814, 9.777757741237360002339327334269

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