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Properties

Label 8-3e4-1.1-c14e4-0-0
Degree $8$
Conductor $81$
Sign $1$
Analytic cond. $193.541$
Root an. cond. $1.93128$
Motivic weight $14$
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  + 2.19e3·3-s + 1.31e4·4-s + 8.25e5·7-s + 1.59e6·9-s + 2.88e7·12-s + 2.01e8·13-s + 2.03e8·16-s + 1.31e9·19-s + 1.81e9·21-s + 8.76e9·25-s + 6.93e9·27-s + 1.08e10·28-s − 3.49e10·31-s + 2.09e10·36-s + 5.55e10·37-s + 4.42e11·39-s − 3.23e11·43-s + 4.47e11·48-s − 2.13e12·49-s + 2.64e12·52-s + 2.88e12·57-s − 4.17e12·61-s + 1.31e12·63-s + 6.60e12·64-s − 1.09e13·67-s − 4.46e13·73-s + 1.92e13·75-s + ⋯
L(s)  = 1  + 1.00·3-s + 0.800·4-s + 1.00·7-s + 0.334·9-s + 0.804·12-s + 3.21·13-s + 0.758·16-s + 1.47·19-s + 1.00·21-s + 1.43·25-s + 0.662·27-s + 0.802·28-s − 1.27·31-s + 0.267·36-s + 0.585·37-s + 3.22·39-s − 1.19·43-s + 0.761·48-s − 3.15·49-s + 2.57·52-s + 1.47·57-s − 1.32·61-s + 0.335·63-s + 1.50·64-s − 1.80·67-s − 4.04·73-s + 1.44·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+7)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $1$
Analytic conductor: \(193.541\)
Root analytic conductor: \(1.93128\)
Motivic weight: \(14\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 81,\ (\ :7, 7, 7, 7),\ 1)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(6.712531293\)
\(L(\frac12)\) \(\approx\) \(6.712531293\)
\(L(8)\) 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$D_{4}$ \( 1 - 244 p^{2} T + 1474 p^{7} T^{2} - 244 p^{16} T^{3} + p^{28} T^{4} \)
good2$C_2^2 \wr C_2$ \( 1 - 205 p^{6} T^{2} - 15369 p^{11} T^{4} - 205 p^{34} T^{6} + p^{56} T^{8} \)
5$C_2^2 \wr C_2$ \( 1 - 1753157972 p T^{2} + 12993265388837262 p^{5} T^{4} - 1753157972 p^{29} T^{6} + p^{56} T^{8} \)
7$D_{4}$ \( ( 1 - 58972 p T + 189295846314 p T^{2} - 58972 p^{15} T^{3} + p^{28} T^{4} )^{2} \)
11$C_2^2 \wr C_2$ \( 1 - 10073573939524 p^{2} T^{2} + \)\(44\!\cdots\!26\)\( p^{4} T^{4} - 10073573939524 p^{30} T^{6} + p^{56} T^{8} \)
13$D_{4}$ \( ( 1 - 7757572 p T + 59987978953062 p^{2} T^{2} - 7757572 p^{15} T^{3} + p^{28} T^{4} )^{2} \)
17$C_2^2 \wr C_2$ \( 1 - 127136234878459780 T^{2} + \)\(29\!\cdots\!18\)\( T^{4} - 127136234878459780 p^{28} T^{6} + p^{56} T^{8} \)
19$D_{4}$ \( ( 1 - 657473620 T + 1701983228472310038 T^{2} - 657473620 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 - 32366964844766210500 T^{2} + \)\(48\!\cdots\!78\)\( T^{4} - 32366964844766210500 p^{28} T^{6} + p^{56} T^{8} \)
29$C_2^2 \wr C_2$ \( 1 - 47831478366103027684 T^{2} + \)\(17\!\cdots\!86\)\( T^{4} - 47831478366103027684 p^{28} T^{6} + p^{56} T^{8} \)
31$D_{4}$ \( ( 1 + 17485355036 T + \)\(10\!\cdots\!66\)\( T^{2} + 17485355036 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 - 27788394964 T + \)\(14\!\cdots\!78\)\( T^{2} - 27788394964 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 - \)\(12\!\cdots\!84\)\( T^{2} + \)\(70\!\cdots\!06\)\( T^{4} - \)\(12\!\cdots\!84\)\( p^{28} T^{6} + p^{56} T^{8} \)
43$D_{4}$ \( ( 1 + 161839464524 T + \)\(12\!\cdots\!98\)\( T^{2} + 161839464524 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
47$C_2^2 \wr C_2$ \( 1 - \)\(62\!\cdots\!00\)\( T^{2} + \)\(20\!\cdots\!38\)\( T^{4} - \)\(62\!\cdots\!00\)\( p^{28} T^{6} + p^{56} T^{8} \)
53$C_2^2 \wr C_2$ \( 1 - \)\(46\!\cdots\!00\)\( T^{2} + \)\(32\!\cdots\!82\)\( p^{2} T^{4} - \)\(46\!\cdots\!00\)\( p^{28} T^{6} + p^{56} T^{8} \)
59$C_2^2 \wr C_2$ \( 1 - \)\(12\!\cdots\!84\)\( T^{2} + \)\(11\!\cdots\!06\)\( T^{4} - \)\(12\!\cdots\!84\)\( p^{28} T^{6} + p^{56} T^{8} \)
61$D_{4}$ \( ( 1 + 2085820813196 T + \)\(10\!\cdots\!86\)\( T^{2} + 2085820813196 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 5482119968876 T + \)\(80\!\cdots\!58\)\( T^{2} + 5482119968876 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
71$C_2^2 \wr C_2$ \( 1 - \)\(10\!\cdots\!84\)\( T^{2} + \)\(83\!\cdots\!86\)\( T^{4} - \)\(10\!\cdots\!84\)\( p^{28} T^{6} + p^{56} T^{8} \)
73$D_{4}$ \( ( 1 + 22322065461404 T + \)\(36\!\cdots\!18\)\( T^{2} + 22322065461404 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 20607721289380 T + \)\(36\!\cdots\!38\)\( T^{2} - 20607721289380 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
83$C_2^2 \wr C_2$ \( 1 - \)\(24\!\cdots\!00\)\( T^{2} + \)\(26\!\cdots\!78\)\( T^{4} - \)\(24\!\cdots\!00\)\( p^{28} T^{6} + p^{56} T^{8} \)
89$C_2^2 \wr C_2$ \( 1 - \)\(68\!\cdots\!04\)\( T^{2} + \)\(19\!\cdots\!66\)\( T^{4} - \)\(68\!\cdots\!04\)\( p^{28} T^{6} + p^{56} T^{8} \)
97$D_{4}$ \( ( 1 - 35264990307844 T + \)\(84\!\cdots\!38\)\( T^{2} - 35264990307844 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
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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

−17.25753596520518826083979307961, −16.34289105597545499246895964263, −16.10481448291544310408982676059, −16.03987864387926298373883356313, −15.05483191703219212729316129938, −14.74763793859724009735873919564, −14.29779699307371061375631772287, −13.72001322629477878391981229248, −13.20507597558205478365606358569, −12.76560516203029275033105041276, −11.63792977795387934645811014307, −11.42137507976158597699455246577, −10.84905188888106928659462293200, −10.31822612198311257777724444974, −9.192216863625731928744722271051, −8.690865224156601003655496090772, −8.165945495526480065346542857443, −7.56689236609126929954637885763, −6.56545316065626169403218547247, −5.88458104682680655033135563454, −4.79063690443691312602537021462, −3.43469876460585567545733956904, −3.12161474398950416375773634813, −1.49790629659222650011613731182, −1.31128672141184471137906988743, 1.31128672141184471137906988743, 1.49790629659222650011613731182, 3.12161474398950416375773634813, 3.43469876460585567545733956904, 4.79063690443691312602537021462, 5.88458104682680655033135563454, 6.56545316065626169403218547247, 7.56689236609126929954637885763, 8.165945495526480065346542857443, 8.690865224156601003655496090772, 9.192216863625731928744722271051, 10.31822612198311257777724444974, 10.84905188888106928659462293200, 11.42137507976158597699455246577, 11.63792977795387934645811014307, 12.76560516203029275033105041276, 13.20507597558205478365606358569, 13.72001322629477878391981229248, 14.29779699307371061375631772287, 14.74763793859724009735873919564, 15.05483191703219212729316129938, 16.03987864387926298373883356313, 16.10481448291544310408982676059, 16.34289105597545499246895964263, 17.25753596520518826083979307961

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