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Properties

Label 8-72e4-1.1-c8e4-0-2
Degree $8$
Conductor $26873856$
Sign $1$
Analytic cond. $740155.$
Root an. cond. $5.41583$
Motivic weight $8$
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  − 2.35e3·7-s + 2.94e4·13-s + 1.14e5·19-s + 1.26e6·25-s + 3.29e6·31-s + 8.87e6·37-s + 1.24e7·43-s + 8.39e6·49-s + 4.19e7·61-s − 3.64e7·67-s + 4.26e7·73-s − 1.76e8·79-s − 6.92e7·91-s − 1.51e8·97-s − 4.65e8·103-s + 1.44e8·109-s + 5.87e8·121-s + 127-s + 131-s − 2.68e8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 0.979·7-s + 1.03·13-s + 0.877·19-s + 3.24·25-s + 3.57·31-s + 4.73·37-s + 3.64·43-s + 1.45·49-s + 3.02·61-s − 1.81·67-s + 1.50·73-s − 4.52·79-s − 1.00·91-s − 1.71·97-s − 4.13·103-s + 1.02·109-s + 2.74·121-s − 0.859·133-s − 1.65·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(26873856\)    =    \(2^{12} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(740155.\)
Root analytic conductor: \(5.41583\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 26873856,\ (\ :4, 4, 4, 4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(10.04609921\)
\(L(\frac12)\) \(\approx\) \(10.04609921\)
\(L(5)\) 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 \( 1 \)
3 \( 1 \)
good5$D_4\times C_2$ \( 1 - 1269408 T^{2} + 1129459106 p^{4} T^{4} - 1269408 p^{16} T^{6} + p^{32} T^{8} \)
7$D_{4}$ \( ( 1 + 24 p^{2} T - 43342 p^{2} T^{2} + 24 p^{10} T^{3} + p^{16} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 587406660 T^{2} + 1444987947609782 p^{2} T^{4} - 587406660 p^{16} T^{6} + p^{32} T^{8} \)
13$D_{4}$ \( ( 1 - 14720 T + 999674946 T^{2} - 14720 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 26231766912 T^{2} + \)\(26\!\cdots\!22\)\( T^{4} - 26231766912 p^{16} T^{6} + p^{32} T^{8} \)
19$D_{4}$ \( ( 1 - 57184 T + 31694540610 T^{2} - 57184 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 233423182020 T^{2} + 47248210025828474918 p^{2} T^{4} - 233423182020 p^{16} T^{6} + p^{32} T^{8} \)
29$D_4\times C_2$ \( 1 - 607262408224 T^{2} + \)\(19\!\cdots\!42\)\( T^{4} - 607262408224 p^{16} T^{6} + p^{32} T^{8} \)
31$D_{4}$ \( ( 1 - 1649608 T + 2158255723794 T^{2} - 1649608 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 - 4438188 T + 11360204509862 T^{2} - 4438188 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 + 9258816255488 T^{2} + \)\(90\!\cdots\!22\)\( T^{4} + 9258816255488 p^{16} T^{6} + p^{32} T^{8} \)
43$D_{4}$ \( ( 1 - 6225488 T + 31705341682434 T^{2} - 6225488 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 8911933394884 T^{2} + \)\(27\!\cdots\!62\)\( T^{4} - 8911933394884 p^{16} T^{6} + p^{32} T^{8} \)
53$D_4\times C_2$ \( 1 - 3995502586656 T^{2} + \)\(32\!\cdots\!90\)\( T^{4} - 3995502586656 p^{16} T^{6} + p^{32} T^{8} \)
59$D_4\times C_2$ \( 1 - 89304151746180 T^{2} - \)\(14\!\cdots\!18\)\( T^{4} - 89304151746180 p^{16} T^{6} + p^{32} T^{8} \)
61$D_{4}$ \( ( 1 - 20976404 T + 335318961538470 T^{2} - 20976404 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 18246608 T + 870988550361474 T^{2} + 18246608 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 1694590269664452 T^{2} + \)\(14\!\cdots\!02\)\( T^{4} - 1694590269664452 p^{16} T^{6} + p^{32} T^{8} \)
73$D_{4}$ \( ( 1 - 21329696 T + 1245748008649602 T^{2} - 21329696 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 88083976 T + 4725843321794322 T^{2} + 88083976 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 7432832941914180 T^{2} + \)\(23\!\cdots\!62\)\( T^{4} - 7432832941914180 p^{16} T^{6} + p^{32} T^{8} \)
89$D_4\times C_2$ \( 1 - 983748845012224 T^{2} + \)\(29\!\cdots\!66\)\( T^{4} - 983748845012224 p^{16} T^{6} + p^{32} T^{8} \)
97$D_{4}$ \( ( 1 + 75913664 T + 6314526498142722 T^{2} + 75913664 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
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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

−9.373540506095147999299611458526, −8.722296186027155277278279104787, −8.313358581381313069982014073412, −8.284825215580577769048828255721, −8.212123533312777011743476162551, −7.36200725907593932137247036381, −7.13802291625827969369077073010, −7.03812934473090463004799299689, −6.59910445223453607670890517413, −6.04785963947403663310221645794, −5.98359452447384909610501197658, −5.79874071214445847650433416231, −5.29313028035118468965427256840, −4.48192605163939637863758331849, −4.42301813071197688561501433223, −4.35370460378776119505061176595, −3.75342499829949379492669045774, −2.93640892340103773992897246634, −2.82544858071761809451759753511, −2.78359583088636175313468536362, −2.33319637318024036052073920190, −1.19070374327182342629272070218, −0.978184553731423007419734098252, −0.877560004736225364598394413141, −0.57225761583781289294760645753, 0.57225761583781289294760645753, 0.877560004736225364598394413141, 0.978184553731423007419734098252, 1.19070374327182342629272070218, 2.33319637318024036052073920190, 2.78359583088636175313468536362, 2.82544858071761809451759753511, 2.93640892340103773992897246634, 3.75342499829949379492669045774, 4.35370460378776119505061176595, 4.42301813071197688561501433223, 4.48192605163939637863758331849, 5.29313028035118468965427256840, 5.79874071214445847650433416231, 5.98359452447384909610501197658, 6.04785963947403663310221645794, 6.59910445223453607670890517413, 7.03812934473090463004799299689, 7.13802291625827969369077073010, 7.36200725907593932137247036381, 8.212123533312777011743476162551, 8.284825215580577769048828255721, 8.313358581381313069982014073412, 8.722296186027155277278279104787, 9.373540506095147999299611458526

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