Properties

Degree $8$
Conductor $7471182096$
Sign $1$
Motivic weight $2$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s − 2·4-s − 12·5-s + 21·9-s + 12·11-s − 12·12-s − 72·15-s + 12·17-s − 12·19-s + 24·20-s − 12·23-s + 40·25-s + 54·27-s + 120·29-s + 72·31-s + 72·33-s − 42·36-s + 40·37-s − 128·43-s − 24·44-s − 252·45-s + 168·47-s + 72·51-s + 108·53-s − 144·55-s − 72·57-s + 168·59-s + ⋯
L(s)  = 1  + 2·3-s − 1/2·4-s − 2.39·5-s + 7/3·9-s + 1.09·11-s − 12-s − 4.79·15-s + 0.705·17-s − 0.631·19-s + 6/5·20-s − 0.521·23-s + 8/5·25-s + 2·27-s + 4.13·29-s + 2.32·31-s + 2.18·33-s − 7/6·36-s + 1.08·37-s − 2.97·43-s − 0.545·44-s − 5.59·45-s + 3.57·47-s + 1.41·51-s + 2.03·53-s − 2.61·55-s − 1.26·57-s + 2.84·59-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Motivic weight: \(2\)
Character: induced by $\chi_{294} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.10112\)
\(L(\frac12)\) \(\approx\) \(4.10112\)
\(L(2)\) 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$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
3$C_2$ \( ( 1 - p T + p T^{2} )^{2} \)
7 \( 1 \)
good5$D_4\times C_2$ \( 1 + 12 T + 104 T^{2} + 672 T^{3} + 3711 T^{4} + 672 p^{2} T^{5} + 104 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \)
11$D_4\times C_2$ \( 1 - 12 T - 116 T^{2} - 216 T^{3} + 35535 T^{4} - 216 p^{2} T^{5} - 116 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \)
13$D_4\times C_2$ \( 1 - 244 T^{2} + 53574 T^{4} - 244 p^{4} T^{6} + p^{8} T^{8} \)
17$C_2^3$ \( 1 - 12 T - 88 T^{2} + 96 p T^{3} - 177 p^{2} T^{4} + 96 p^{3} T^{5} - 88 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \)
19$D_4\times C_2$ \( 1 + 12 T + 398 T^{2} + 4200 T^{3} + 9507 T^{4} + 4200 p^{2} T^{5} + 398 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \)
23$D_4\times C_2$ \( 1 + 12 T - 788 T^{2} - 1512 T^{3} + 512607 T^{4} - 1512 p^{2} T^{5} - 788 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \)
29$C_2$ \( ( 1 - 30 T + p^{2} T^{2} )^{4} \)
31$D_4\times C_2$ \( 1 - 72 T + 3218 T^{2} - 107280 T^{3} + 2957187 T^{4} - 107280 p^{2} T^{5} + 3218 p^{4} T^{6} - 72 p^{6} T^{7} + p^{8} T^{8} \)
37$D_4\times C_2$ \( 1 - 40 T + 1054 T^{2} + 87680 T^{3} - 3766445 T^{4} + 87680 p^{2} T^{5} + 1054 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} \)
41$D_4\times C_2$ \( 1 - 4960 T^{2} + 11110434 T^{4} - 4960 p^{4} T^{6} + p^{8} T^{8} \)
43$D_{4}$ \( ( 1 + 64 T + 2922 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 168 T + 16082 T^{2} - 23856 p T^{3} + 27363 p^{2} T^{4} - 23856 p^{3} T^{5} + 16082 p^{4} T^{6} - 168 p^{6} T^{7} + p^{8} T^{8} \)
53$D_4\times C_2$ \( 1 - 108 T + 3418 T^{2} - 283824 T^{3} + 27341859 T^{4} - 283824 p^{2} T^{5} + 3418 p^{4} T^{6} - 108 p^{6} T^{7} + p^{8} T^{8} \)
59$D_4\times C_2$ \( 1 - 168 T + 16322 T^{2} - 1161552 T^{3} + 68435283 T^{4} - 1161552 p^{2} T^{5} + 16322 p^{4} T^{6} - 168 p^{6} T^{7} + p^{8} T^{8} \)
61$D_4\times C_2$ \( 1 - 24 T + 7586 T^{2} - 177456 T^{3} + 41539827 T^{4} - 177456 p^{2} T^{5} + 7586 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \)
67$D_4\times C_2$ \( 1 + 88 T - 2882 T^{2} + 145024 T^{3} + 57997939 T^{4} + 145024 p^{2} T^{5} - 2882 p^{4} T^{6} + 88 p^{6} T^{7} + p^{8} T^{8} \)
71$D_{4}$ \( ( 1 + 60 T + 4484 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 24 T + 6842 T^{2} + 159600 T^{3} + 16847427 T^{4} + 159600 p^{2} T^{5} + 6842 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \)
79$D_4\times C_2$ \( 1 + 64 T + 958 T^{2} - 598016 T^{3} - 54666173 T^{4} - 598016 p^{2} T^{5} + 958 p^{4} T^{6} + 64 p^{6} T^{7} + p^{8} T^{8} \)
83$D_4\times C_2$ \( 1 - 16612 T^{2} + 134415078 T^{4} - 16612 p^{4} T^{6} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 + 324 T + 56936 T^{2} + 7109856 T^{3} + 695968527 T^{4} + 7109856 p^{2} T^{5} + 56936 p^{4} T^{6} + 324 p^{6} T^{7} + p^{8} T^{8} \)
97$D_4\times C_2$ \( 1 - 24820 T^{2} + 317127462 T^{4} - 24820 p^{4} T^{6} + p^{8} T^{8} \)
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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

−8.350372312551556340697704115193, −8.193330988592548779873497908426, −8.022004803403390064516972442992, −7.76782770921302616225797303082, −7.38984346726022459837412422355, −7.05234682051587544273261600142, −6.90260381356856843820718693944, −6.82542589359470167849095017927, −6.39320548256240311305814639110, −5.97725140538962698289242091342, −5.70143346467320353274644825730, −5.34833976106795719295254590057, −4.56979915207340690639701127954, −4.55537043851791870286105648100, −4.46623690828482854975893324571, −3.99420022922746849639968796605, −3.89061378311851311645173330816, −3.72876978288576317481396792878, −3.05941227294398664619558387192, −2.93733282214066670633627877269, −2.56449419282933204546869393149, −2.32740498244718668295045756806, −1.33236585970647825780478904007, −1.06908117758609312279566108070, −0.50554266829983590438363000804, 0.50554266829983590438363000804, 1.06908117758609312279566108070, 1.33236585970647825780478904007, 2.32740498244718668295045756806, 2.56449419282933204546869393149, 2.93733282214066670633627877269, 3.05941227294398664619558387192, 3.72876978288576317481396792878, 3.89061378311851311645173330816, 3.99420022922746849639968796605, 4.46623690828482854975893324571, 4.55537043851791870286105648100, 4.56979915207340690639701127954, 5.34833976106795719295254590057, 5.70143346467320353274644825730, 5.97725140538962698289242091342, 6.39320548256240311305814639110, 6.82542589359470167849095017927, 6.90260381356856843820718693944, 7.05234682051587544273261600142, 7.38984346726022459837412422355, 7.76782770921302616225797303082, 8.022004803403390064516972442992, 8.193330988592548779873497908426, 8.350372312551556340697704115193

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