Properties

Label 8-14e4-1.1-c6e4-0-0
Degree $8$
Conductor $38416$
Sign $1$
Analytic cond. $107.604$
Root an. cond. $1.79464$
Motivic weight $6$
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  + 64·4-s + 308·7-s + 2.00e3·9-s − 4.44e3·11-s + 3.07e3·16-s + 4.05e4·23-s + 1.09e4·25-s + 1.97e4·28-s − 1.82e4·29-s + 1.28e5·36-s − 2.31e4·37-s − 4.46e4·43-s − 2.84e5·44-s − 1.07e5·49-s + 2.48e5·53-s + 6.17e5·63-s + 1.31e5·64-s − 4.34e5·67-s − 4.51e5·71-s − 1.36e6·77-s + 2.09e6·79-s + 2.14e6·81-s + 2.59e6·92-s − 8.89e6·99-s + 6.97e5·100-s + 6.34e5·107-s + 5.82e6·109-s + ⋯
L(s)  = 1  + 4-s + 0.897·7-s + 2.74·9-s − 3.33·11-s + 3/4·16-s + 3.33·23-s + 0.697·25-s + 0.897·28-s − 0.748·29-s + 2.74·36-s − 0.457·37-s − 0.562·43-s − 3.33·44-s − 0.915·49-s + 1.66·53-s + 2.46·63-s + 1/2·64-s − 1.44·67-s − 1.26·71-s − 2.99·77-s + 4.24·79-s + 4.03·81-s + 3.33·92-s − 9.17·99-s + 0.697·100-s + 0.517·107-s + 4.49·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(38416\)    =    \(2^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(107.604\)
Root analytic conductor: \(1.79464\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 38416,\ (\ :3, 3, 3, 3),\ 1)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.837119394\)
\(L(\frac12)\) \(\approx\) \(3.837119394\)
\(L(4)\) 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$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
7$D_{4}$ \( 1 - 44 p T + 4134 p^{2} T^{2} - 44 p^{7} T^{3} + p^{12} T^{4} \)
good3$C_2^2 \wr C_2$ \( 1 - 668 p T^{2} + 208022 p^{2} T^{4} - 668 p^{13} T^{6} + p^{24} T^{8} \)
5$C_2^2 \wr C_2$ \( 1 - 436 p^{2} T^{2} + 412902 p^{4} T^{4} - 436 p^{14} T^{6} + p^{24} T^{8} \)
11$D_{4}$ \( ( 1 + 2220 T + 4770614 T^{2} + 2220 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
13$C_2^2 \wr C_2$ \( 1 - 3940180 T^{2} + 17707018928070 T^{4} - 3940180 p^{12} T^{6} + p^{24} T^{8} \)
17$C_2^2 \wr C_2$ \( 1 - 56335492 T^{2} + 1681094907776646 T^{4} - 56335492 p^{12} T^{6} + p^{24} T^{8} \)
19$C_2^2 \wr C_2$ \( 1 - 122207956 T^{2} + 7623942006639558 T^{4} - 122207956 p^{12} T^{6} + p^{24} T^{8} \)
23$D_{4}$ \( ( 1 - 20292 T + 396782822 T^{2} - 20292 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 + 9132 T - 168260074 T^{2} + 9132 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
31$C_2^2 \wr C_2$ \( 1 - 1275081988 T^{2} + 1557093819506538246 T^{4} - 1275081988 p^{12} T^{6} + p^{24} T^{8} \)
37$D_{4}$ \( ( 1 + 11596 T + 3283526454 T^{2} + 11596 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 - 9929192068 T^{2} + 52045566850295970246 T^{4} - 9929192068 p^{12} T^{6} + p^{24} T^{8} \)
43$D_{4}$ \( ( 1 + 22348 T + 2121261174 T^{2} + 22348 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
47$C_2^2 \wr C_2$ \( 1 - 22996383364 T^{2} + \)\(27\!\cdots\!18\)\( T^{4} - 22996383364 p^{12} T^{6} + p^{24} T^{8} \)
53$D_{4}$ \( ( 1 - 124308 T + 44342337206 T^{2} - 124308 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
59$C_2^2 \wr C_2$ \( 1 + 11871974636 T^{2} + \)\(33\!\cdots\!86\)\( T^{4} + 11871974636 p^{12} T^{6} + p^{24} T^{8} \)
61$C_2^2 \wr C_2$ \( 1 - 84307090516 T^{2} + \)\(41\!\cdots\!58\)\( T^{4} - 84307090516 p^{12} T^{6} + p^{24} T^{8} \)
67$D_{4}$ \( ( 1 + 217388 T + 192725823126 T^{2} + 217388 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 + 225804 T + 160756795334 T^{2} + 225804 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
73$C_2^2 \wr C_2$ \( 1 - 377326906756 T^{2} + \)\(71\!\cdots\!98\)\( T^{4} - 377326906756 p^{12} T^{6} + p^{24} T^{8} \)
79$D_{4}$ \( ( 1 - 1046452 T + 754787202918 T^{2} - 1046452 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
83$C_2^2 \wr C_2$ \( 1 - 828919590868 T^{2} + \)\(35\!\cdots\!70\)\( T^{4} - 828919590868 p^{12} T^{6} + p^{24} T^{8} \)
89$C_2^2 \wr C_2$ \( 1 - 1729674521860 T^{2} + \)\(12\!\cdots\!30\)\( T^{4} - 1729674521860 p^{12} T^{6} + p^{24} T^{8} \)
97$C_2^2 \wr C_2$ \( 1 - 3289276805380 T^{2} + \)\(40\!\cdots\!70\)\( T^{4} - 3289276805380 p^{12} T^{6} + p^{24} T^{8} \)
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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

−13.56451171486002962272060594041, −13.09515784612772290387638888658, −12.85915816698137519740319576869, −12.61267282436283628360058488894, −12.30240115825826189790306003323, −11.47619625570003954078704736690, −11.04606202617613967104475756779, −10.94433243691035871113541021821, −10.34238995854355790171013150365, −10.24306775010029967900648671490, −9.954250041488454821248368793856, −9.023600513172322870039378606323, −8.743131243383779283035843669582, −7.85411874414407392759396515208, −7.56705771619632141047465505399, −7.39285511371144148459613847782, −6.97470933455247246630931941210, −6.33859245696523188220522060595, −5.21015436235711202081742289833, −5.04012306530305913138641290254, −4.69914401864769701379118414498, −3.44123114552857990266343436475, −2.65164762697882211504464742228, −1.86371346581523819161058772347, −0.961920847447946679456680070997, 0.961920847447946679456680070997, 1.86371346581523819161058772347, 2.65164762697882211504464742228, 3.44123114552857990266343436475, 4.69914401864769701379118414498, 5.04012306530305913138641290254, 5.21015436235711202081742289833, 6.33859245696523188220522060595, 6.97470933455247246630931941210, 7.39285511371144148459613847782, 7.56705771619632141047465505399, 7.85411874414407392759396515208, 8.743131243383779283035843669582, 9.023600513172322870039378606323, 9.954250041488454821248368793856, 10.24306775010029967900648671490, 10.34238995854355790171013150365, 10.94433243691035871113541021821, 11.04606202617613967104475756779, 11.47619625570003954078704736690, 12.30240115825826189790306003323, 12.61267282436283628360058488894, 12.85915816698137519740319576869, 13.09515784612772290387638888658, 13.56451171486002962272060594041

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