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Properties

Label 8-2e32-1.1-c11e4-0-5
Degree $8$
Conductor $4294967296$
Sign $1$
Analytic cond. $1.49685\times 10^{9}$
Root an. cond. $14.0248$
Motivic weight $11$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.10e4·5-s − 1.23e5·9-s + 9.81e5·13-s − 1.54e7·17-s + 1.78e8·25-s − 5.97e8·29-s − 3.53e8·37-s + 1.86e9·41-s + 2.60e9·45-s + 6.48e8·49-s + 7.93e8·53-s + 9.70e8·61-s − 2.06e10·65-s + 4.98e10·73-s − 1.75e10·81-s + 3.25e11·85-s − 2.43e11·89-s + 2.73e11·97-s + 1.17e11·101-s − 2.84e10·109-s + 1.06e10·113-s − 1.21e11·117-s − 1.00e12·121-s − 8.28e11·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 3.01·5-s − 0.698·9-s + 0.732·13-s − 2.64·17-s + 3.64·25-s − 5.41·29-s − 0.838·37-s + 2.50·41-s + 2.10·45-s + 0.327·49-s + 0.260·53-s + 0.147·61-s − 2.20·65-s + 2.81·73-s − 0.559·81-s + 7.96·85-s − 4.62·89-s + 3.23·97-s + 1.11·101-s − 0.177·109-s + 0.0545·113-s − 0.511·117-s − 3.52·121-s − 2.42·125-s + 16.3·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32}\)
Sign: $1$
Analytic conductor: \(1.49685\times 10^{9}\)
Root analytic conductor: \(14.0248\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{32} ,\ ( \ : 11/2, 11/2, 11/2, 11/2 ),\ 1 )\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) 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$C_2^2 \wr C_2$ \( 1 + 41252 p T^{2} + 135246226 p^{5} T^{4} + 41252 p^{23} T^{6} + p^{44} T^{8} \)
5$D_{4}$ \( ( 1 + 2104 p T + 3077338 p^{2} T^{2} + 2104 p^{12} T^{3} + p^{22} T^{4} )^{2} \)
7$C_2^2 \wr C_2$ \( 1 - 648091044 T^{2} - 445304836774055702 p T^{4} - 648091044 p^{22} T^{6} + p^{44} T^{8} \)
11$C_2^2 \wr C_2$ \( 1 + 1005087545548 T^{2} + \)\(41\!\cdots\!42\)\( T^{4} + 1005087545548 p^{22} T^{6} + p^{44} T^{8} \)
13$D_{4}$ \( ( 1 - 490536 T + 2604491994874 T^{2} - 490536 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 + 7741380 T + 81930187981542 T^{2} + 7741380 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
19$C_2^2 \wr C_2$ \( 1 + 237839255219244 T^{2} + \)\(37\!\cdots\!42\)\( T^{4} + 237839255219244 p^{22} T^{6} + p^{44} T^{8} \)
23$C_2^2 \wr C_2$ \( 1 + 1718730072212764 T^{2} + \)\(14\!\cdots\!98\)\( T^{4} + 1718730072212764 p^{22} T^{6} + p^{44} T^{8} \)
29$D_{4}$ \( ( 1 + 298998616 T + 45563363921451866 T^{2} + 298998616 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
31$C_2^2 \wr C_2$ \( 1 + 60557373408056700 T^{2} + \)\(18\!\cdots\!66\)\( T^{4} + 60557373408056700 p^{22} T^{6} + p^{44} T^{8} \)
37$D_{4}$ \( ( 1 + 176935352 T + 182817422840092298 T^{2} + 176935352 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 930586700 T + 1314791334596820118 T^{2} - 930586700 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
43$C_2^2 \wr C_2$ \( 1 + 19392974369474124 T^{2} - \)\(92\!\cdots\!94\)\( T^{4} + 19392974369474124 p^{22} T^{6} + p^{44} T^{8} \)
47$C_2^2 \wr C_2$ \( 1 - 2057165629748089796 T^{2} + \)\(11\!\cdots\!06\)\( T^{4} - 2057165629748089796 p^{22} T^{6} + p^{44} T^{8} \)
53$D_{4}$ \( ( 1 - 396653672 T - 1223548603505340310 T^{2} - 396653672 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
59$C_2^2 \wr C_2$ \( 1 + \)\(10\!\cdots\!24\)\( T^{2} + \)\(42\!\cdots\!82\)\( T^{4} + \)\(10\!\cdots\!24\)\( p^{22} T^{6} + p^{44} T^{8} \)
61$D_{4}$ \( ( 1 - 485489928 T + 86556209220015576218 T^{2} - 485489928 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
67$C_2^2 \wr C_2$ \( 1 + \)\(46\!\cdots\!96\)\( T^{2} + \)\(83\!\cdots\!06\)\( T^{4} + \)\(46\!\cdots\!96\)\( p^{22} T^{6} + p^{44} T^{8} \)
71$C_2^2 \wr C_2$ \( 1 - \)\(16\!\cdots\!60\)\( T^{2} + \)\(11\!\cdots\!06\)\( T^{4} - \)\(16\!\cdots\!60\)\( p^{22} T^{6} + p^{44} T^{8} \)
73$D_{4}$ \( ( 1 - 24947808660 T + \)\(74\!\cdots\!90\)\( T^{2} - 24947808660 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
79$C_2^2 \wr C_2$ \( 1 + \)\(18\!\cdots\!72\)\( T^{2} + \)\(19\!\cdots\!02\)\( T^{4} + \)\(18\!\cdots\!72\)\( p^{22} T^{6} + p^{44} T^{8} \)
83$C_2^2 \wr C_2$ \( 1 + \)\(22\!\cdots\!96\)\( T^{2} + \)\(42\!\cdots\!18\)\( T^{4} + \)\(22\!\cdots\!96\)\( p^{22} T^{6} + p^{44} T^{8} \)
89$D_{4}$ \( ( 1 + 121791685388 T + \)\(91\!\cdots\!50\)\( T^{2} + 121791685388 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 136670791132 T + \)\(13\!\cdots\!26\)\( T^{2} - 136670791132 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
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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

−7.45272229951438602308629386768, −7.28935117443127694538986276159, −6.86764738765985418839626396826, −6.59718250408551469537559113821, −6.43912639077261957049274550582, −5.90508167937851004028900969296, −5.78085850615880573833123698443, −5.62050339827531676057506869538, −5.20512613815232158216770197789, −4.89706770103359309004886953209, −4.59840728044474776963381614911, −4.30398894326298705776324897931, −4.04885122459005984576827267072, −3.79481406949913210109855943797, −3.71386436931235298415144859992, −3.53628850819387869793351674523, −3.45581357882929784243969256874, −2.59735018216116972080323646931, −2.51184597460634647328311993754, −2.35007361637705283391938097947, −1.99431626764712452799828689908, −1.62524390952369746824133610495, −1.19920575567112397613576036341, −1.03005220481908019931913427414, −0.50792440704267187784165607453, 0, 0, 0, 0, 0.50792440704267187784165607453, 1.03005220481908019931913427414, 1.19920575567112397613576036341, 1.62524390952369746824133610495, 1.99431626764712452799828689908, 2.35007361637705283391938097947, 2.51184597460634647328311993754, 2.59735018216116972080323646931, 3.45581357882929784243969256874, 3.53628850819387869793351674523, 3.71386436931235298415144859992, 3.79481406949913210109855943797, 4.04885122459005984576827267072, 4.30398894326298705776324897931, 4.59840728044474776963381614911, 4.89706770103359309004886953209, 5.20512613815232158216770197789, 5.62050339827531676057506869538, 5.78085850615880573833123698443, 5.90508167937851004028900969296, 6.43912639077261957049274550582, 6.59718250408551469537559113821, 6.86764738765985418839626396826, 7.28935117443127694538986276159, 7.45272229951438602308629386768

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