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Properties

Label 8-2e20-1.1-c15e4-0-0
Degree $8$
Conductor $1048576$
Sign $1$
Analytic cond. $4.34727\times 10^{6}$
Root an. cond. $6.75736$
Motivic weight $15$
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  + 3.78e5·5-s − 1.69e7·9-s + 4.19e8·13-s + 5.91e8·17-s + 5.02e10·25-s + 9.12e10·29-s + 7.82e11·37-s + 1.39e12·41-s − 6.41e12·45-s + 1.45e12·49-s + 2.74e13·53-s + 9.98e13·61-s + 1.58e14·65-s − 1.07e14·73-s − 3.68e13·81-s + 2.24e14·85-s − 9.70e14·89-s − 4.18e15·97-s + 1.38e15·101-s − 9.69e14·109-s − 7.10e15·113-s − 7.11e15·117-s − 1.50e16·121-s + 6.20e15·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 2.16·5-s − 1.18·9-s + 1.85·13-s + 0.349·17-s + 1.64·25-s + 0.982·29-s + 1.35·37-s + 1.11·41-s − 2.56·45-s + 0.305·49-s + 3.20·53-s + 4.06·61-s + 4.01·65-s − 1.13·73-s − 0.179·81-s + 0.757·85-s − 2.32·89-s − 5.25·97-s + 1.28·101-s − 0.508·109-s − 2.84·113-s − 2.19·117-s − 3.59·121-s + 1.16·125-s + 2.12·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s+15/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(1048576\)    =    \(2^{20}\)
Sign: $1$
Analytic conductor: \(4.34727\times 10^{6}\)
Root analytic conductor: \(6.75736\)
Motivic weight: \(15\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 1048576,\ (\ :15/2, 15/2, 15/2, 15/2),\ 1)\)

Particular Values

\(L(8)\) \(\approx\) \(4.568634203\)
\(L(\frac12)\) \(\approx\) \(4.568634203\)
\(L(\frac{17}{2})\) 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 \( 1 \)
good3$C_2^2 \wr C_2$ \( 1 + 69796 p^{5} T^{2} + 1335526728658 p^{5} T^{4} + 69796 p^{35} T^{6} + p^{60} T^{8} \)
5$D_{4}$ \( ( 1 - 37852 p T + 1143373934 p^{2} T^{2} - 37852 p^{16} T^{3} + p^{30} T^{4} )^{2} \)
7$C_2^2 \wr C_2$ \( 1 - 207337852604 p T^{2} + \)\(11\!\cdots\!58\)\( p^{3} T^{4} - 207337852604 p^{31} T^{6} + p^{60} T^{8} \)
11$C_2^2 \wr C_2$ \( 1 + 15027641705873804 T^{2} + \)\(75\!\cdots\!86\)\( p^{2} T^{4} + 15027641705873804 p^{30} T^{6} + p^{60} T^{8} \)
13$D_{4}$ \( ( 1 - 16133292 p T + 371665207193822 p^{2} T^{2} - 16133292 p^{16} T^{3} + p^{30} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 - 295917508 T + 5661117481739052902 T^{2} - 295917508 p^{15} T^{3} + p^{30} T^{4} )^{2} \)
19$C_2^2 \wr C_2$ \( 1 + 13726564111136812396 T^{2} + \)\(88\!\cdots\!06\)\( T^{4} + 13726564111136812396 p^{30} T^{6} + p^{60} T^{8} \)
23$C_2^2 \wr C_2$ \( 1 + 95722800574964483228 T^{2} + \)\(23\!\cdots\!94\)\( T^{4} + 95722800574964483228 p^{30} T^{6} + p^{60} T^{8} \)
29$D_{4}$ \( ( 1 - 45619152220 T + \)\(17\!\cdots\!98\)\( T^{2} - 45619152220 p^{15} T^{3} + p^{30} T^{4} )^{2} \)
31$C_2^2 \wr C_2$ \( 1 + \)\(54\!\cdots\!04\)\( T^{2} + \)\(17\!\cdots\!06\)\( T^{4} + \)\(54\!\cdots\!04\)\( p^{30} T^{6} + p^{60} T^{8} \)
37$D_{4}$ \( ( 1 - 391226640332 T + \)\(69\!\cdots\!42\)\( T^{2} - 391226640332 p^{15} T^{3} + p^{30} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 695508799220 T + \)\(26\!\cdots\!02\)\( T^{2} - 695508799220 p^{15} T^{3} + p^{30} T^{4} )^{2} \)
43$C_2^2 \wr C_2$ \( 1 + \)\(11\!\cdots\!28\)\( T^{2} + \)\(54\!\cdots\!94\)\( T^{4} + \)\(11\!\cdots\!28\)\( p^{30} T^{6} + p^{60} T^{8} \)
47$C_2^2 \wr C_2$ \( 1 + \)\(18\!\cdots\!72\)\( T^{2} + \)\(26\!\cdots\!94\)\( T^{4} + \)\(18\!\cdots\!72\)\( p^{30} T^{6} + p^{60} T^{8} \)
53$D_{4}$ \( ( 1 - 13710604776364 T + \)\(14\!\cdots\!38\)\( T^{2} - 13710604776364 p^{15} T^{3} + p^{30} T^{4} )^{2} \)
59$C_2^2 \wr C_2$ \( 1 + \)\(12\!\cdots\!96\)\( T^{2} + \)\(63\!\cdots\!06\)\( T^{4} + \)\(12\!\cdots\!96\)\( p^{30} T^{6} + p^{60} T^{8} \)
61$D_{4}$ \( ( 1 - 49940568380380 T + \)\(16\!\cdots\!02\)\( T^{2} - 49940568380380 p^{15} T^{3} + p^{30} T^{4} )^{2} \)
67$C_2^2 \wr C_2$ \( 1 + \)\(82\!\cdots\!72\)\( T^{2} + \)\(71\!\cdots\!94\)\( T^{4} + \)\(82\!\cdots\!72\)\( p^{30} T^{6} + p^{60} T^{8} \)
71$C_2^2 \wr C_2$ \( 1 + \)\(10\!\cdots\!04\)\( T^{2} + \)\(94\!\cdots\!06\)\( T^{4} + \)\(10\!\cdots\!04\)\( p^{30} T^{6} + p^{60} T^{8} \)
73$D_{4}$ \( ( 1 + 53601560152236 T + \)\(14\!\cdots\!38\)\( T^{2} + 53601560152236 p^{15} T^{3} + p^{30} T^{4} )^{2} \)
79$C_2^2 \wr C_2$ \( 1 + \)\(25\!\cdots\!96\)\( T^{2} + \)\(15\!\cdots\!06\)\( T^{4} + \)\(25\!\cdots\!96\)\( p^{30} T^{6} + p^{60} T^{8} \)
83$C_2^2 \wr C_2$ \( 1 + \)\(23\!\cdots\!28\)\( T^{2} + \)\(21\!\cdots\!94\)\( T^{4} + \)\(23\!\cdots\!28\)\( p^{30} T^{6} + p^{60} T^{8} \)
89$D_{4}$ \( ( 1 + 485365640175820 T + \)\(19\!\cdots\!98\)\( T^{2} + 485365640175820 p^{15} T^{3} + p^{30} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 2091084224632668 T + \)\(22\!\cdots\!42\)\( T^{2} + 2091084224632668 p^{15} T^{3} + p^{30} T^{4} )^{2} \)
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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

−9.406835855392783714373741053805, −8.832548608024304545555037005521, −8.386178190153422782290298691761, −8.298490684827248844331962144500, −8.255683349143794865938913828518, −7.37981489367896883650033060056, −6.91467028761653264678787007899, −6.76822703382298078701117968659, −6.34547401514804715141681133017, −5.74502848027610153718619736395, −5.70042470992069949220438134717, −5.66595194014617017570824960290, −5.32962996102068298878373654536, −4.58426779174670274183358915594, −4.05145445362851103696175730521, −3.92873588961967973087349051123, −3.43532217312714933784985807926, −2.65252431470071012217599982671, −2.55771290572372417412884753681, −2.50833515403247027245766666923, −1.86846986736552603554994429709, −1.28221855739760611674964585835, −1.05474467637054563241661327541, −0.973772409920768027263188547750, −0.21626326753537711194708386769, 0.21626326753537711194708386769, 0.973772409920768027263188547750, 1.05474467637054563241661327541, 1.28221855739760611674964585835, 1.86846986736552603554994429709, 2.50833515403247027245766666923, 2.55771290572372417412884753681, 2.65252431470071012217599982671, 3.43532217312714933784985807926, 3.92873588961967973087349051123, 4.05145445362851103696175730521, 4.58426779174670274183358915594, 5.32962996102068298878373654536, 5.66595194014617017570824960290, 5.70042470992069949220438134717, 5.74502848027610153718619736395, 6.34547401514804715141681133017, 6.76822703382298078701117968659, 6.91467028761653264678787007899, 7.37981489367896883650033060056, 8.255683349143794865938913828518, 8.298490684827248844331962144500, 8.386178190153422782290298691761, 8.832548608024304545555037005521, 9.406835855392783714373741053805

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