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Properties

Label 8-18e4-1.1-c20e4-0-1
Degree $8$
Conductor $104976$
Sign $1$
Analytic cond. $4.33606\times 10^{6}$
Root an. cond. $6.75518$
Motivic weight $20$
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  − 1.04e6·4-s + 1.96e8·7-s + 3.11e11·13-s + 8.24e11·16-s − 1.07e13·19-s + 4.56e13·25-s − 2.06e14·28-s − 1.26e15·31-s − 1.41e16·37-s + 6.81e15·43-s − 2.94e17·49-s − 3.26e17·52-s − 3.35e18·61-s − 5.76e17·64-s − 5.95e18·67-s − 2.53e18·73-s + 1.13e19·76-s + 3.25e19·79-s + 6.12e19·91-s − 4.01e19·97-s − 4.78e19·100-s + 1.06e20·103-s − 6.68e20·109-s + 1.62e20·112-s − 6.11e20·121-s + 1.32e21·124-s + 127-s + ⋯
L(s)  = 1  − 4-s + 0.696·7-s + 2.25·13-s + 3/4·16-s − 1.75·19-s + 0.478·25-s − 0.696·28-s − 1.54·31-s − 2.93·37-s + 0.315·43-s − 3.69·49-s − 2.25·52-s − 4.70·61-s − 1/2·64-s − 3.26·67-s − 0.588·73-s + 1.75·76-s + 3.44·79-s + 1.57·91-s − 0.544·97-s − 0.478·100-s + 0.792·103-s − 2.82·109-s + 0.522·112-s − 0.909·121-s + 1.54·124-s − 1.22·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+10)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(104976\)    =    \(2^{4} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(4.33606\times 10^{6}\)
Root analytic conductor: \(6.75518\)
Motivic weight: \(20\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 104976,\ (\ :10, 10, 10, 10),\ 1)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(0.8007985130\)
\(L(\frac12)\) \(\approx\) \(0.8007985130\)
\(L(11)\) 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$C_2$ \( ( 1 + p^{19} T^{2} )^{2} \)
3 \( 1 \)
good5$D_4\times C_2$ \( 1 - 1824479501728 p^{2} T^{2} + \)\(11\!\cdots\!66\)\( p^{6} T^{4} - 1824479501728 p^{42} T^{6} + p^{80} T^{8} \)
7$D_{4}$ \( ( 1 - 98306344 T + 3305995185513714 p^{2} T^{2} - 98306344 p^{20} T^{3} + p^{40} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 + 5055212790746814812 p^{2} T^{2} + \)\(28\!\cdots\!58\)\( p^{4} T^{4} + 5055212790746814812 p^{42} T^{6} + p^{80} T^{8} \)
13$D_{4}$ \( ( 1 - 155722461568 T + \)\(32\!\cdots\!66\)\( p T^{2} - 155722461568 p^{20} T^{3} + p^{40} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - \)\(45\!\cdots\!92\)\( p^{2} T^{2} + \)\(89\!\cdots\!78\)\( p^{4} T^{4} - \)\(45\!\cdots\!92\)\( p^{42} T^{6} + p^{80} T^{8} \)
19$D_{4}$ \( ( 1 + 14945222048 p^{2} T + \)\(12\!\cdots\!18\)\( p^{2} T^{2} + 14945222048 p^{22} T^{3} + p^{40} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - \)\(17\!\cdots\!60\)\( T^{2} + \)\(56\!\cdots\!02\)\( T^{4} - \)\(17\!\cdots\!60\)\( p^{40} T^{6} + p^{80} T^{8} \)
29$D_4\times C_2$ \( 1 - \)\(11\!\cdots\!08\)\( T^{2} + \)\(10\!\cdots\!18\)\( T^{4} - \)\(11\!\cdots\!08\)\( p^{40} T^{6} + p^{80} T^{8} \)
31$D_{4}$ \( ( 1 + 632677815055352 T + \)\(14\!\cdots\!78\)\( T^{2} + 632677815055352 p^{20} T^{3} + p^{40} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 7058600422494836 T + \)\(29\!\cdots\!26\)\( T^{2} + 7058600422494836 p^{20} T^{3} + p^{40} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - \)\(20\!\cdots\!28\)\( T^{2} + \)\(26\!\cdots\!98\)\( T^{4} - \)\(20\!\cdots\!28\)\( p^{40} T^{6} + p^{80} T^{8} \)
43$D_{4}$ \( ( 1 - 3405868285710160 T + \)\(54\!\cdots\!02\)\( T^{2} - 3405868285710160 p^{20} T^{3} + p^{40} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - \)\(22\!\cdots\!04\)\( T^{2} - \)\(99\!\cdots\!94\)\( T^{4} - \)\(22\!\cdots\!04\)\( p^{40} T^{6} + p^{80} T^{8} \)
53$D_4\times C_2$ \( 1 - \)\(70\!\cdots\!80\)\( T^{2} + \)\(31\!\cdots\!02\)\( T^{4} - \)\(70\!\cdots\!80\)\( p^{40} T^{6} + p^{80} T^{8} \)
59$D_4\times C_2$ \( 1 - \)\(64\!\cdots\!20\)\( T^{2} + \)\(23\!\cdots\!02\)\( T^{4} - \)\(64\!\cdots\!20\)\( p^{40} T^{6} + p^{80} T^{8} \)
61$D_{4}$ \( ( 1 + 1678072979377090700 T + \)\(16\!\cdots\!02\)\( T^{2} + 1678072979377090700 p^{20} T^{3} + p^{40} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 2975687854427895248 T + \)\(58\!\cdots\!78\)\( T^{2} + 2975687854427895248 p^{20} T^{3} + p^{40} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - \)\(28\!\cdots\!80\)\( T^{2} + \)\(41\!\cdots\!02\)\( T^{4} - \)\(28\!\cdots\!80\)\( p^{40} T^{6} + p^{80} T^{8} \)
73$D_{4}$ \( ( 1 + 1265103510102984416 T - \)\(68\!\cdots\!34\)\( T^{2} + 1265103510102984416 p^{20} T^{3} + p^{40} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 16293592190260912312 T + \)\(21\!\cdots\!38\)\( T^{2} - 16293592190260912312 p^{20} T^{3} + p^{40} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - \)\(12\!\cdots\!88\)\( T^{2} + \)\(10\!\cdots\!38\)\( T^{4} - \)\(12\!\cdots\!88\)\( p^{40} T^{6} + p^{80} T^{8} \)
89$D_4\times C_2$ \( 1 - \)\(40\!\cdots\!60\)\( T^{2} + \)\(19\!\cdots\!02\)\( T^{4} - \)\(40\!\cdots\!60\)\( p^{40} T^{6} + p^{80} T^{8} \)
97$D_{4}$ \( ( 1 + 20069467303393313216 T + \)\(15\!\cdots\!66\)\( T^{2} + 20069467303393313216 p^{20} T^{3} + p^{40} T^{4} )^{2} \)
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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

−9.425071081172359903857941690195, −9.042135229317940191117177837735, −8.757030150063769969998459466664, −8.406751764102347277266966806948, −8.250466974782827518371825970182, −7.67840741533197424425918608141, −7.54701379359538235882603646368, −6.80476732096243531249300620578, −6.34754630058214972243610710866, −6.29223481387188659528266826712, −5.86951560091625310423264751918, −5.16418483236551459898323836013, −5.14626727052566296551342858152, −4.56068831405778698751561415608, −4.33328335304425003521009508555, −3.83632616482752657182727773258, −3.55528310388930255379229290155, −3.08562177668764440120434891506, −2.94565464967557238181280609565, −1.84621570145090849754838095563, −1.78817519034486164281519785789, −1.42918961119265534808353770001, −1.29022870492456051046436922320, −0.38696713697652577078568935723, −0.18552888898871297704143680311, 0.18552888898871297704143680311, 0.38696713697652577078568935723, 1.29022870492456051046436922320, 1.42918961119265534808353770001, 1.78817519034486164281519785789, 1.84621570145090849754838095563, 2.94565464967557238181280609565, 3.08562177668764440120434891506, 3.55528310388930255379229290155, 3.83632616482752657182727773258, 4.33328335304425003521009508555, 4.56068831405778698751561415608, 5.14626727052566296551342858152, 5.16418483236551459898323836013, 5.86951560091625310423264751918, 6.29223481387188659528266826712, 6.34754630058214972243610710866, 6.80476732096243531249300620578, 7.54701379359538235882603646368, 7.67840741533197424425918608141, 8.250466974782827518371825970182, 8.406751764102347277266966806948, 8.757030150063769969998459466664, 9.042135229317940191117177837735, 9.425071081172359903857941690195

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