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Properties

Label 8-3e12-1.1-c14e4-0-1
Degree $8$
Conductor $531441$
Sign $1$
Analytic cond. $1.26982\times 10^{6}$
Root an. cond. $5.79386$
Motivic weight $14$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.15e4·4-s + 4.48e5·7-s + 6.64e7·13-s + 1.02e8·16-s − 1.38e9·19-s + 1.06e10·25-s + 9.66e9·28-s − 6.31e9·31-s − 4.09e11·37-s − 7.62e11·43-s − 2.35e12·49-s + 1.43e12·52-s + 6.85e11·61-s + 2.13e11·64-s + 1.40e13·67-s + 2.21e13·73-s − 2.98e13·76-s + 6.86e12·79-s + 2.98e13·91-s − 2.43e14·97-s + 2.29e14·100-s − 1.34e14·103-s + 3.05e13·109-s + 4.59e13·112-s + 1.40e15·121-s − 1.35e14·124-s + 127-s + ⋯
L(s)  = 1  + 1.31·4-s + 0.545·7-s + 1.05·13-s + 0.381·16-s − 1.54·19-s + 1.75·25-s + 0.716·28-s − 0.229·31-s − 4.31·37-s − 2.80·43-s − 3.47·49-s + 1.39·52-s + 0.218·61-s + 0.0484·64-s + 2.31·67-s + 2.00·73-s − 2.03·76-s + 0.357·79-s + 0.577·91-s − 3.01·97-s + 2.29·100-s − 1.09·103-s + 0.167·109-s + 0.207·112-s + 3.68·121-s − 0.301·124-s − 0.844·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531441 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531441 ^{s/2} \, \Gamma_{\C}(s+7)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(531441\)    =    \(3^{12}\)
Sign: $1$
Analytic conductor: \(1.26982\times 10^{6}\)
Root analytic conductor: \(5.79386\)
Motivic weight: \(14\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 531441,\ (\ :7, 7, 7, 7),\ 1)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(6.704475909\)
\(L(\frac12)\) \(\approx\) \(6.704475909\)
\(L(8)\) 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 \( 1 \)
good2$C_2^2 \wr C_2$ \( 1 - 10763 p T^{2} + 5639829 p^{6} T^{4} - 10763 p^{29} T^{6} + p^{56} T^{8} \)
5$C_2^2 \wr C_2$ \( 1 - 2136415268 p T^{2} + 779973890947507902 p^{3} T^{4} - 2136415268 p^{29} T^{6} + p^{56} T^{8} \)
7$D_{4}$ \( ( 1 - 32062 p T + 178904741541 p T^{2} - 32062 p^{15} T^{3} + p^{28} T^{4} )^{2} \)
11$C_2^2 \wr C_2$ \( 1 - 127372736726684 p T^{2} + \)\(58\!\cdots\!26\)\( p^{3} T^{4} - 127372736726684 p^{29} T^{6} + p^{56} T^{8} \)
13$D_{4}$ \( ( 1 - 2557246 p T + 31980957884691 p^{2} T^{2} - 2557246 p^{15} T^{3} + p^{28} T^{4} )^{2} \)
17$C_2^2 \wr C_2$ \( 1 - 204648251906308756 T^{2} + \)\(23\!\cdots\!66\)\( T^{4} - 204648251906308756 p^{28} T^{6} + p^{56} T^{8} \)
19$D_{4}$ \( ( 1 + 692686598 T + 990189953395989243 T^{2} + 692686598 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 - 6247560894653514196 T^{2} + \)\(33\!\cdots\!66\)\( T^{4} - 6247560894653514196 p^{28} T^{6} + p^{56} T^{8} \)
29$C_2^2 \wr C_2$ \( 1 - \)\(34\!\cdots\!24\)\( T^{2} + \)\(20\!\cdots\!66\)\( T^{4} - \)\(34\!\cdots\!24\)\( p^{28} T^{6} + p^{56} T^{8} \)
31$D_{4}$ \( ( 1 + 3155494772 T - \)\(10\!\cdots\!62\)\( T^{2} + 3155494772 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 204629173466 T + \)\(23\!\cdots\!67\)\( T^{2} + 204629173466 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 - \)\(56\!\cdots\!44\)\( T^{2} + \)\(14\!\cdots\!26\)\( T^{4} - \)\(56\!\cdots\!44\)\( p^{28} T^{6} + p^{56} T^{8} \)
43$D_{4}$ \( ( 1 + 381441255524 T + \)\(12\!\cdots\!42\)\( T^{2} + 381441255524 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
47$C_2^2 \wr C_2$ \( 1 - \)\(10\!\cdots\!16\)\( T^{2} + \)\(38\!\cdots\!86\)\( T^{4} - \)\(10\!\cdots\!16\)\( p^{28} T^{6} + p^{56} T^{8} \)
53$C_2^2 \wr C_2$ \( 1 + \)\(40\!\cdots\!64\)\( T^{2} + \)\(76\!\cdots\!46\)\( T^{4} + \)\(40\!\cdots\!64\)\( p^{28} T^{6} + p^{56} T^{8} \)
59$C_2^2 \wr C_2$ \( 1 - \)\(21\!\cdots\!04\)\( T^{2} + \)\(19\!\cdots\!46\)\( T^{4} - \)\(21\!\cdots\!04\)\( p^{28} T^{6} + p^{56} T^{8} \)
61$D_{4}$ \( ( 1 - 342878160358 T + \)\(17\!\cdots\!23\)\( T^{2} - 342878160358 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 - 7017151026922 T + \)\(85\!\cdots\!79\)\( T^{2} - 7017151026922 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
71$C_2^2 \wr C_2$ \( 1 - \)\(17\!\cdots\!64\)\( T^{2} + \)\(20\!\cdots\!46\)\( T^{4} - \)\(17\!\cdots\!64\)\( p^{28} T^{6} + p^{56} T^{8} \)
73$D_{4}$ \( ( 1 - 11098744983886 T + \)\(15\!\cdots\!67\)\( T^{2} - 11098744983886 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 3434320563922 T + \)\(46\!\cdots\!83\)\( T^{2} - 3434320563922 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
83$C_2^2 \wr C_2$ \( 1 - \)\(10\!\cdots\!16\)\( T^{2} + \)\(11\!\cdots\!46\)\( T^{4} - \)\(10\!\cdots\!16\)\( p^{28} T^{6} + p^{56} T^{8} \)
89$C_2^2 \wr C_2$ \( 1 - \)\(86\!\cdots\!04\)\( T^{2} + \)\(73\!\cdots\!66\)\( T^{4} - \)\(86\!\cdots\!04\)\( p^{28} T^{6} + p^{56} T^{8} \)
97$D_{4}$ \( ( 1 + 121773619060946 T + \)\(14\!\cdots\!67\)\( T^{2} + 121773619060946 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
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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

−9.674218244256886003108810796880, −9.673330935928662647405118253912, −8.710444942367302712121627358016, −8.564795962751385917745591393716, −8.314653869722091109380520577017, −8.278089891500644689839457637623, −7.57052957276382955578116273899, −6.90363132603266501546393967507, −6.66727627281360357325577099703, −6.64101169731220395374741062366, −6.59249427128767193145692946314, −5.52329553186400640106126602255, −5.44267854887156965292637676894, −5.02349429110505446761466380083, −4.54233262796765001584589712095, −4.17670027732078644464582615456, −3.31964935216139471836199231909, −3.29518960340929097363840867223, −3.13371934686053897206560121690, −2.03523267117628102918366145156, −2.01232974651216085067637829258, −1.72324537101499645735644323422, −1.41449003820266548618067860770, −0.49738787547196007130855852066, −0.45618296817073361982064137311, 0.45618296817073361982064137311, 0.49738787547196007130855852066, 1.41449003820266548618067860770, 1.72324537101499645735644323422, 2.01232974651216085067637829258, 2.03523267117628102918366145156, 3.13371934686053897206560121690, 3.29518960340929097363840867223, 3.31964935216139471836199231909, 4.17670027732078644464582615456, 4.54233262796765001584589712095, 5.02349429110505446761466380083, 5.44267854887156965292637676894, 5.52329553186400640106126602255, 6.59249427128767193145692946314, 6.64101169731220395374741062366, 6.66727627281360357325577099703, 6.90363132603266501546393967507, 7.57052957276382955578116273899, 8.278089891500644689839457637623, 8.314653869722091109380520577017, 8.564795962751385917745591393716, 8.710444942367302712121627358016, 9.673330935928662647405118253912, 9.674218244256886003108810796880

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