Properties

Degree $8$
Conductor $1.216\times 10^{12}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 9-s − 12·11-s − 8·19-s + 24·29-s − 10·31-s + 36-s − 12·41-s − 12·44-s − 11·49-s + 12·59-s − 4·61-s − 64-s − 36·71-s − 8·76-s − 14·79-s + 6·89-s − 12·99-s − 24·101-s − 8·109-s + 24·116-s + 58·121-s − 10·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1/2·4-s + 1/3·9-s − 3.61·11-s − 1.83·19-s + 4.45·29-s − 1.79·31-s + 1/6·36-s − 1.87·41-s − 1.80·44-s − 1.57·49-s + 1.56·59-s − 0.512·61-s − 1/8·64-s − 4.27·71-s − 0.917·76-s − 1.57·79-s + 0.635·89-s − 1.20·99-s − 2.38·101-s − 0.766·109-s + 2.22·116-s + 5.27·121-s − 0.898·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1050} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1400143473\)
\(L(\frac12)\) \(\approx\) \(0.1400143473\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
3$C_2^2$ \( 1 - T^{2} + T^{4} \)
5 \( 1 \)
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
good11$C_2^2$ \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \)
17$C_2^3$ \( 1 + 25 T^{2} + 336 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^3$ \( 1 + 10 T^{2} - 1269 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 + 13 T^{2} - 2040 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 38 T^{2} - 1365 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 + 70 T^{2} + 411 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2$ \( ( 1 + 9 T + p T^{2} )^{4} \)
73$C_2^3$ \( 1 - 50 T^{2} - 2829 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 + 7 T - 30 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 95 T^{2} + p^{2} T^{4} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.05113462997955719860660766105, −6.91921657920594762036504799848, −6.67495449620542958797644509197, −6.48450996864042061856084324001, −6.19804490651383943895751881021, −6.00719064690910479443635281134, −5.68359393025157227187238677789, −5.54145640256597988457700014120, −5.27368255830283830124067904262, −4.91346294357125680424989067728, −4.80957545152105573906385653746, −4.74490855693351358488029120181, −4.40322265965327691128932082319, −4.06153056014524679663152706536, −3.88629644280544625628283227217, −3.37970273041077530920096457158, −2.91113373488978393947408426507, −2.90069226179787753120502043436, −2.74456253382226711071637230133, −2.60013347672798769382334623438, −2.00394974214287043147633418900, −1.73424659765548099039396300791, −1.56315211787416114318088498282, −0.77563940836790596139122098489, −0.094496227362436737330275604808, 0.094496227362436737330275604808, 0.77563940836790596139122098489, 1.56315211787416114318088498282, 1.73424659765548099039396300791, 2.00394974214287043147633418900, 2.60013347672798769382334623438, 2.74456253382226711071637230133, 2.90069226179787753120502043436, 2.91113373488978393947408426507, 3.37970273041077530920096457158, 3.88629644280544625628283227217, 4.06153056014524679663152706536, 4.40322265965327691128932082319, 4.74490855693351358488029120181, 4.80957545152105573906385653746, 4.91346294357125680424989067728, 5.27368255830283830124067904262, 5.54145640256597988457700014120, 5.68359393025157227187238677789, 6.00719064690910479443635281134, 6.19804490651383943895751881021, 6.48450996864042061856084324001, 6.67495449620542958797644509197, 6.91921657920594762036504799848, 7.05113462997955719860660766105

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