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Properties

Label 8-546e4-1.1-c3e4-0-0
Degree $8$
Conductor $88873149456$
Sign $1$
Analytic cond. $1.07704\times 10^{6}$
Root an. cond. $5.67582$
Motivic weight $3$
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  − 4·2-s + 6·3-s + 4·4-s + 6·5-s − 24·6-s − 20·7-s + 16·8-s + 9·9-s − 24·10-s − 36·11-s + 24·12-s + 52·13-s + 80·14-s + 36·15-s − 64·16-s − 120·17-s − 36·18-s − 118·19-s + 24·20-s − 120·21-s + 144·22-s + 42·23-s + 96·24-s + 244·25-s − 208·26-s − 54·27-s − 80·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 1/2·4-s + 0.536·5-s − 1.63·6-s − 1.07·7-s + 0.707·8-s + 1/3·9-s − 0.758·10-s − 0.986·11-s + 0.577·12-s + 1.10·13-s + 1.52·14-s + 0.619·15-s − 16-s − 1.71·17-s − 0.471·18-s − 1.42·19-s + 0.268·20-s − 1.24·21-s + 1.39·22-s + 0.380·23-s + 0.816·24-s + 1.95·25-s − 1.56·26-s − 0.384·27-s − 0.539·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(1.07704\times 10^{6}\)
Root analytic conductor: \(5.67582\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.01387158678\)
\(L(\frac12)\) \(\approx\) \(0.01387158678\)
\(L(\frac{5}{2})\) 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$C_2$ \( ( 1 + p T + p^{2} T^{2} )^{2} \)
3$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
7$C_2^2$ \( 1 + 20 T + 93 p T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \)
13$C_1$ \( ( 1 - p T )^{4} \)
good5$D_4\times C_2$ \( 1 - 6 T - 208 T^{2} + 36 T^{3} + 39411 T^{4} + 36 p^{3} T^{5} - 208 p^{6} T^{6} - 6 p^{9} T^{7} + p^{12} T^{8} \)
11$D_4\times C_2$ \( 1 + 36 T - 1315 T^{2} - 1836 T^{3} + 3320784 T^{4} - 1836 p^{3} T^{5} - 1315 p^{6} T^{6} + 36 p^{9} T^{7} + p^{12} T^{8} \)
17$D_4\times C_2$ \( 1 + 120 T + 2189 T^{2} + 286200 T^{3} + 54223752 T^{4} + 286200 p^{3} T^{5} + 2189 p^{6} T^{6} + 120 p^{9} T^{7} + p^{12} T^{8} \)
19$D_4\times C_2$ \( 1 + 118 T - 1115 T^{2} + 155878 T^{3} + 83620924 T^{4} + 155878 p^{3} T^{5} - 1115 p^{6} T^{6} + 118 p^{9} T^{7} + p^{12} T^{8} \)
23$D_4\times C_2$ \( 1 - 42 T - 19636 T^{2} + 123228 T^{3} + 288461523 T^{4} + 123228 p^{3} T^{5} - 19636 p^{6} T^{6} - 42 p^{9} T^{7} + p^{12} T^{8} \)
29$D_{4}$ \( ( 1 + 54 T + 48547 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 28 T - 15254 T^{2} - 1219232 T^{3} - 653642381 T^{4} - 1219232 p^{3} T^{5} - 15254 p^{6} T^{6} + 28 p^{9} T^{7} + p^{12} T^{8} \)
37$D_4\times C_2$ \( 1 + 154 T - 72584 T^{2} - 770924 T^{3} + 5506677043 T^{4} - 770924 p^{3} T^{5} - 72584 p^{6} T^{6} + 154 p^{9} T^{7} + p^{12} T^{8} \)
41$D_{4}$ \( ( 1 - 546 T + 193996 T^{2} - 546 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 200 T + 12954 T^{2} + 200 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 822 T + 302057 T^{2} + 136436382 T^{3} + 58666379868 T^{4} + 136436382 p^{3} T^{5} + 302057 p^{6} T^{6} + 822 p^{9} T^{7} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 - 90 T - 190819 T^{2} + 8895150 T^{3} + 16198503732 T^{4} + 8895150 p^{3} T^{5} - 190819 p^{6} T^{6} - 90 p^{9} T^{7} + p^{12} T^{8} \)
59$D_4\times C_2$ \( 1 + 600 T - 140623 T^{2} + 53919000 T^{3} + 134506414488 T^{4} + 53919000 p^{3} T^{5} - 140623 p^{6} T^{6} + 600 p^{9} T^{7} + p^{12} T^{8} \)
61$D_4\times C_2$ \( 1 - 44 T - 403775 T^{2} + 34804 p T^{3} + 30271864 p^{2} T^{4} + 34804 p^{4} T^{5} - 403775 p^{6} T^{6} - 44 p^{9} T^{7} + p^{12} T^{8} \)
67$D_4\times C_2$ \( 1 - 50 T - 545651 T^{2} + 2668750 T^{3} + 209259229132 T^{4} + 2668750 p^{3} T^{5} - 545651 p^{6} T^{6} - 50 p^{9} T^{7} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 + 66 T + 705151 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 70 T - 689984 T^{2} - 5820500 T^{3} + 329623363867 T^{4} - 5820500 p^{3} T^{5} - 689984 p^{6} T^{6} + 70 p^{9} T^{7} + p^{12} T^{8} \)
79$D_4\times C_2$ \( 1 - 1004 T + 80974 T^{2} + 59272144 T^{3} + 119328213619 T^{4} + 59272144 p^{3} T^{5} + 80974 p^{6} T^{6} - 1004 p^{9} T^{7} + p^{12} T^{8} \)
83$D_{4}$ \( ( 1 + 816 T + 1115098 T^{2} + 816 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 282 T + 195320 T^{2} + 430256988 T^{3} - 539634532101 T^{4} + 430256988 p^{3} T^{5} + 195320 p^{6} T^{6} - 282 p^{9} T^{7} + p^{12} T^{8} \)
97$D_{4}$ \( ( 1 - 1486 T + 2026260 T^{2} - 1486 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
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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

−7.63895585955155448924392158714, −7.06160685122303986387054534099, −7.05632132405676577197226228694, −6.55537984449411162856489134116, −6.50876875996685753366836484407, −6.25271310595312942006673861860, −6.23752279434939814634398025677, −5.60827157513150352206445087079, −5.54400052877536735166770770749, −5.00794801752937970532170675883, −4.81297090560770657818078459884, −4.69332465179365634390097575260, −4.14512384704835108592570632680, −4.04503673473321020757465147777, −3.76443068629937468415470493447, −3.23387360586726140856530069563, −2.97546712142729630976411463028, −2.88761711938818788871091813228, −2.52777409527052387920842458033, −2.06140689327859446780712541897, −1.86181347333188988082796800062, −1.55151364233386196288286391606, −0.996986323588825374815426125722, −0.57928545840376412565383915652, −0.02584923422611186985743498030, 0.02584923422611186985743498030, 0.57928545840376412565383915652, 0.996986323588825374815426125722, 1.55151364233386196288286391606, 1.86181347333188988082796800062, 2.06140689327859446780712541897, 2.52777409527052387920842458033, 2.88761711938818788871091813228, 2.97546712142729630976411463028, 3.23387360586726140856530069563, 3.76443068629937468415470493447, 4.04503673473321020757465147777, 4.14512384704835108592570632680, 4.69332465179365634390097575260, 4.81297090560770657818078459884, 5.00794801752937970532170675883, 5.54400052877536735166770770749, 5.60827157513150352206445087079, 6.23752279434939814634398025677, 6.25271310595312942006673861860, 6.50876875996685753366836484407, 6.55537984449411162856489134116, 7.05632132405676577197226228694, 7.06160685122303986387054534099, 7.63895585955155448924392158714

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