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Properties

Label 8-7623e4-1.1-c1e4-0-3
Degree $8$
Conductor $3.377\times 10^{15}$
Sign $1$
Analytic cond. $1.37281\times 10^{7}$
Root an. cond. $7.80192$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 6·5-s + 4·7-s − 8-s + 12·10-s + 8·14-s − 16-s − 3·17-s − 3·19-s + 8·23-s + 8·25-s + 3·29-s − 3·31-s − 10·32-s − 6·34-s + 24·35-s − 7·37-s − 6·38-s − 6·40-s + 4·41-s − 8·43-s + 16·46-s + 14·47-s + 10·49-s + 16·50-s + 9·53-s − 4·56-s + ⋯
L(s)  = 1  + 1.41·2-s + 2.68·5-s + 1.51·7-s − 0.353·8-s + 3.79·10-s + 2.13·14-s − 1/4·16-s − 0.727·17-s − 0.688·19-s + 1.66·23-s + 8/5·25-s + 0.557·29-s − 0.538·31-s − 1.76·32-s − 1.02·34-s + 4.05·35-s − 1.15·37-s − 0.973·38-s − 0.948·40-s + 0.624·41-s − 1.21·43-s + 2.35·46-s + 2.04·47-s + 10/7·49-s + 2.26·50-s + 1.23·53-s − 0.534·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 7^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(1.37281\times 10^{7}\)
Root analytic conductor: \(7.80192\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 7^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(29.21845740\)
\(L(\frac12)\) \(\approx\) \(29.21845740\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7$C_1$ \( ( 1 - T )^{4} \)
11 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 - p T + p^{2} T^{2} - 7 T^{3} + 13 T^{4} - 7 p T^{5} + p^{4} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 28 T^{2} - 87 T^{3} + 229 T^{4} - 87 p T^{5} + 28 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 20 T^{2} + 5 p T^{3} + 153 T^{4} + 5 p^{2} T^{5} + 20 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 45 T^{2} + 62 T^{3} + 881 T^{4} + 62 p T^{5} + 45 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 47 T^{2} + 36 T^{3} + 919 T^{4} + 36 p T^{5} + 47 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 83 T^{2} - 402 T^{3} + 2555 T^{4} - 402 p T^{5} + 83 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 62 T^{2} - 9 p T^{3} + 2319 T^{4} - 9 p^{2} T^{5} + 62 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 100 T^{2} + 299 T^{3} + 4283 T^{4} + 299 p T^{5} + 100 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 154 T^{2} + 767 T^{3} + 8653 T^{4} + 767 p T^{5} + 154 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 107 T^{2} - 470 T^{3} + 5491 T^{4} - 470 p T^{5} + 107 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
43$D_{4}$ \( ( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 162 T^{2} - 1449 T^{3} + 10505 T^{4} - 1449 p T^{5} + 162 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 9 T + 108 T^{2} - 725 T^{3} + 4961 T^{4} - 725 p T^{5} + 108 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 25 T + 428 T^{2} - 4725 T^{3} + 42353 T^{4} - 4725 p T^{5} + 428 p^{2} T^{6} - 25 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 19 T + 238 T^{2} - 2447 T^{3} + 20599 T^{4} - 2447 p T^{5} + 238 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 5 p T^{2} + 3060 T^{3} + 35713 T^{4} + 3060 p T^{5} + 5 p^{3} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 201 T^{2} - 812 T^{3} + 17469 T^{4} - 812 p T^{5} + 201 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 176 T^{2} - 1649 T^{3} + 20013 T^{4} - 1649 p T^{5} + 176 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 308 T^{2} + 1735 T^{3} + 35911 T^{4} + 1735 p T^{5} + 308 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + T + 96 T^{2} + 1149 T^{3} + 4403 T^{4} + 1149 p T^{5} + 96 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 17 T + 312 T^{2} - 3419 T^{3} + 38939 T^{4} - 3419 p T^{5} + 312 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 278 T^{2} + 2415 T^{3} + 30889 T^{4} + 2415 p T^{5} + 278 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.54550132698682753606864793233, −5.12806556694172756667426358838, −5.09081051342264791213032339358, −5.08793363740680971380090778588, −5.01991139124635812587809989127, −4.65816056758299019399978944769, −4.37218214300969828365020130897, −4.27666582054712226197239796238, −4.21576505546798691881247878998, −3.82087333240394094427056364142, −3.69615846170419875331401073204, −3.63924377940535851629912829210, −3.40557179011451806418045050134, −2.96531603108225015284273822220, −2.61591356765712113452246364481, −2.42843299167210931775736701033, −2.39452752985246507291431743720, −2.19059011191706472212842807519, −1.99629286529619532527689231395, −1.80836306475757154826217348687, −1.50212543390285906280860303624, −1.36505537872671501774385865563, −0.978645046279921661832347036148, −0.63627559022666469651044724620, −0.40450543110189062076705325119, 0.40450543110189062076705325119, 0.63627559022666469651044724620, 0.978645046279921661832347036148, 1.36505537872671501774385865563, 1.50212543390285906280860303624, 1.80836306475757154826217348687, 1.99629286529619532527689231395, 2.19059011191706472212842807519, 2.39452752985246507291431743720, 2.42843299167210931775736701033, 2.61591356765712113452246364481, 2.96531603108225015284273822220, 3.40557179011451806418045050134, 3.63924377940535851629912829210, 3.69615846170419875331401073204, 3.82087333240394094427056364142, 4.21576505546798691881247878998, 4.27666582054712226197239796238, 4.37218214300969828365020130897, 4.65816056758299019399978944769, 5.01991139124635812587809989127, 5.08793363740680971380090778588, 5.09081051342264791213032339358, 5.12806556694172756667426358838, 5.54550132698682753606864793233

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