Properties

Label 8-1950e4-1.1-c1e4-0-26
Degree $8$
Conductor $1.446\times 10^{13}$
Sign $1$
Analytic cond. $58782.3$
Root an. cond. $3.94598$
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  + 2·3-s + 4-s − 6·7-s + 9-s − 6·11-s + 2·12-s + 8·17-s + 6·19-s − 12·21-s + 2·23-s − 2·27-s − 6·28-s + 2·29-s − 12·33-s + 36-s + 12·37-s + 36·41-s + 2·43-s − 6·44-s + 8·49-s + 16·51-s + 12·53-s + 12·57-s + 8·61-s − 6·63-s − 64-s + 42·67-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/2·4-s − 2.26·7-s + 1/3·9-s − 1.80·11-s + 0.577·12-s + 1.94·17-s + 1.37·19-s − 2.61·21-s + 0.417·23-s − 0.384·27-s − 1.13·28-s + 0.371·29-s − 2.08·33-s + 1/6·36-s + 1.97·37-s + 5.62·41-s + 0.304·43-s − 0.904·44-s + 8/7·49-s + 2.24·51-s + 1.64·53-s + 1.58·57-s + 1.02·61-s − 0.755·63-s − 1/8·64-s + 5.13·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(58782.3\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1950} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.474892911\)
\(L(\frac12)\) \(\approx\) \(4.474892911\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - T + T^{2} )^{2} \)
5 \( 1 \)
13$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
good7$D_4\times C_2$ \( 1 + 6 T + 4 p T^{2} + 96 T^{3} + 291 T^{4} + 96 p T^{5} + 4 p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 + 6 T + 28 T^{2} + 96 T^{3} + 267 T^{4} + 96 p T^{5} + 28 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2$$\times$$C_2^2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \)
19$D_4\times C_2$ \( 1 - 6 T + 44 T^{2} - 192 T^{3} + 891 T^{4} - 192 p T^{5} + 44 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 2 T - 16 T^{2} + 52 T^{3} - 221 T^{4} + 52 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 - 2 T - 43 T^{2} + 22 T^{3} + 1252 T^{4} + 22 p T^{5} - 43 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 92 T^{2} + 3846 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + 48 p T^{4} - 12 p^{2} T^{5} + 85 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 - 36 T + 621 T^{2} - 6804 T^{3} + 51752 T^{4} - 6804 p T^{5} + 621 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 - 2 T - 8 T^{2} + 148 T^{3} - 1877 T^{4} + 148 p T^{5} - 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 116 T^{2} + 6810 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 - 6 T + 103 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^3$ \( 1 + 54 T^{2} - 565 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 8 T - 47 T^{2} + 88 T^{3} + 4696 T^{4} + 88 p T^{5} - 47 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 42 T + 868 T^{2} - 11760 T^{3} + 113307 T^{4} - 11760 p T^{5} + 868 p^{2} T^{6} - 42 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 6 T + 148 T^{2} - 816 T^{3} + 14307 T^{4} - 816 p T^{5} + 148 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 158 T^{2} + 16131 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 228 T^{2} + 24074 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 + 12 T + 202 T^{2} + 1848 T^{3} + 20067 T^{4} + 1848 p T^{5} + 202 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2^3$ \( 1 + 158 T^{2} + 15555 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.67106704244770423842933119036, −6.34677380575521993574943930300, −5.93878941861164290816487413109, −5.76908142816943838228611766825, −5.75631243232626555413781224685, −5.67480403986000409170212217760, −5.32376443054217428520377202082, −5.20613980985200007037442651216, −4.93636391552560156415034608678, −4.24680890881607784523993358335, −4.24515032070989903854532850534, −4.24027703156403770087322553962, −3.82833653798591209144892445658, −3.48812757672656145818868770554, −3.37421997135436850283389383238, −2.97652263717091875013959214311, −2.95367496971954104060929628859, −2.67676563492584604856053156722, −2.46784770139233999295301256328, −2.38255743788492795765107849161, −2.19564666206182251916709746146, −1.24489510305999793323676883842, −1.09577675853303949248725458752, −0.876542969013062183631029622083, −0.37364629654869154881557387663, 0.37364629654869154881557387663, 0.876542969013062183631029622083, 1.09577675853303949248725458752, 1.24489510305999793323676883842, 2.19564666206182251916709746146, 2.38255743788492795765107849161, 2.46784770139233999295301256328, 2.67676563492584604856053156722, 2.95367496971954104060929628859, 2.97652263717091875013959214311, 3.37421997135436850283389383238, 3.48812757672656145818868770554, 3.82833653798591209144892445658, 4.24027703156403770087322553962, 4.24515032070989903854532850534, 4.24680890881607784523993358335, 4.93636391552560156415034608678, 5.20613980985200007037442651216, 5.32376443054217428520377202082, 5.67480403986000409170212217760, 5.75631243232626555413781224685, 5.76908142816943838228611766825, 5.93878941861164290816487413109, 6.34677380575521993574943930300, 6.67106704244770423842933119036

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