Properties

Label 8-1950e4-1.1-c1e4-0-42
Degree $8$
Conductor $1.446\times 10^{13}$
Sign $1$
Analytic cond. $58782.3$
Root an. cond. $3.94598$
Motivic weight $1$
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  − 2·2-s + 4-s + 6·7-s + 2·8-s + 9-s − 6·11-s + 14·13-s − 12·14-s − 4·16-s − 2·18-s − 6·19-s + 12·22-s + 12·23-s − 28·26-s + 6·28-s − 4·29-s + 2·32-s + 36-s − 2·37-s + 12·38-s − 6·44-s − 24·46-s + 12·47-s + 23·49-s + 14·52-s + 12·56-s + 8·58-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s + 2.26·7-s + 0.707·8-s + 1/3·9-s − 1.80·11-s + 3.88·13-s − 3.20·14-s − 16-s − 0.471·18-s − 1.37·19-s + 2.55·22-s + 2.50·23-s − 5.49·26-s + 1.13·28-s − 0.742·29-s + 0.353·32-s + 1/6·36-s − 0.328·37-s + 1.94·38-s − 0.904·44-s − 3.53·46-s + 1.75·47-s + 23/7·49-s + 1.94·52-s + 1.60·56-s + 1.05·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(58782.3\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.834298045\)
\(L(\frac12)\) \(\approx\) \(3.834298045\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T + T^{2} )^{2} \)
3$C_2^2$ \( 1 - T^{2} + T^{4} \)
5 \( 1 \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
good7$C_2^2$ \( ( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 + 6 T + 3 p T^{2} + 126 T^{3} + 452 T^{4} + 126 p T^{5} + 3 p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2^3$ \( 1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 + 6 T + 37 T^{2} + 150 T^{3} + 492 T^{4} + 150 p T^{5} + 37 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
29$D_4\times C_2$ \( 1 + 4 T - 34 T^{2} - 32 T^{3} + 1195 T^{4} - 32 p T^{5} - 34 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 20 T^{2} + 1254 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 + 2 T - 23 T^{2} - 94 T^{3} - 788 T^{4} - 94 p T^{5} - 23 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^3$ \( 1 + 66 T^{2} + 2675 T^{4} + 66 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2^3$ \( 1 + 50 T^{2} + 651 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} \)
47$D_{4}$ \( ( 1 - 6 T + 91 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 198 T^{2} + 15371 T^{4} - 198 p^{2} T^{6} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 12 T + 114 T^{2} - 792 T^{3} + 3707 T^{4} - 792 p T^{5} + 114 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 8 T - 62 T^{2} - 32 T^{3} + 8251 T^{4} - 32 p T^{5} - 62 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 4 T - 110 T^{2} + 32 T^{3} + 10315 T^{4} + 32 p T^{5} - 110 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 + 24 T + 346 T^{2} + 3696 T^{3} + 32307 T^{4} + 3696 p T^{5} + 346 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2^2$ \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 20 T + 210 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_{4}$ \( ( 1 - 12 T + 190 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 42 T + 909 T^{2} - 13482 T^{3} + 147452 T^{4} - 13482 p T^{5} + 909 p^{2} T^{6} - 42 p^{3} T^{7} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 + 4 T - 74 T^{2} - 416 T^{3} - 2861 T^{4} - 416 p T^{5} - 74 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.69671762119779998915690477180, −6.30081815158833371619385505464, −6.10982331571858888452229545531, −5.82089796822931547569123152291, −5.64588348886244351261702141529, −5.55901273552213636163081027197, −5.38497579962527784224650990777, −4.93686525618766859552425449750, −4.82176699759887838096354829144, −4.64255641108054443677641647082, −4.45968776680412345934241876520, −4.12490396768012320428632664951, −3.92272466743445211565200754301, −3.60301788456713829409251235302, −3.59369203730034410161808166629, −3.15990392642374962054066467665, −2.70281238004129357633447137931, −2.65513609387938025199748331114, −2.19344963941081354114626516443, −1.84641394623897468406935452787, −1.70999028271412598695349522804, −1.41800325894065019969365434806, −1.08076511406329344796919726155, −0.70118150265477416887726142717, −0.61284921550639710534244728043, 0.61284921550639710534244728043, 0.70118150265477416887726142717, 1.08076511406329344796919726155, 1.41800325894065019969365434806, 1.70999028271412598695349522804, 1.84641394623897468406935452787, 2.19344963941081354114626516443, 2.65513609387938025199748331114, 2.70281238004129357633447137931, 3.15990392642374962054066467665, 3.59369203730034410161808166629, 3.60301788456713829409251235302, 3.92272466743445211565200754301, 4.12490396768012320428632664951, 4.45968776680412345934241876520, 4.64255641108054443677641647082, 4.82176699759887838096354829144, 4.93686525618766859552425449750, 5.38497579962527784224650990777, 5.55901273552213636163081027197, 5.64588348886244351261702141529, 5.82089796822931547569123152291, 6.10982331571858888452229545531, 6.30081815158833371619385505464, 6.69671762119779998915690477180

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