Properties

Label 8-320e4-1.1-c7e4-0-0
Degree $8$
Conductor $10485760000$
Sign $1$
Analytic cond. $9.98529\times 10^{7}$
Root an. cond. $9.99816$
Motivic weight $7$
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  + 500·5-s − 3.07e3·9-s − 712·13-s + 3.86e4·17-s + 1.56e5·25-s + 3.23e5·29-s + 9.30e5·37-s − 2.49e5·41-s − 1.53e6·45-s − 2.11e6·49-s + 2.09e6·53-s + 4.15e6·61-s − 3.56e5·65-s − 1.09e7·73-s − 5.30e5·81-s + 1.93e7·85-s − 1.91e7·89-s + 1.15e7·97-s + 5.16e7·101-s + 1.14e7·109-s + 2.66e7·113-s + 2.19e6·117-s − 1.49e7·121-s + 3.90e7·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1.78·5-s − 1.40·9-s − 0.0898·13-s + 1.90·17-s + 2·25-s + 2.46·29-s + 3.02·37-s − 0.564·41-s − 2.51·45-s − 2.56·49-s + 1.93·53-s + 2.34·61-s − 0.160·65-s − 3.29·73-s − 0.110·81-s + 3.40·85-s − 2.88·89-s + 1.28·97-s + 4.98·101-s + 0.849·109-s + 1.73·113-s + 0.126·117-s − 0.769·121-s + 1.78·125-s + 4.40·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(9.98529\times 10^{7}\)
Root analytic conductor: \(9.99816\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 5^{4} ,\ ( \ : 7/2, 7/2, 7/2, 7/2 ),\ 1 )\)

Particular Values

\(L(4)\) \(\approx\) \(0.1848653456\)
\(L(\frac12)\) \(\approx\) \(0.1848653456\)
\(L(\frac{9}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - p^{3} T )^{4} \)
good3$C_2^2 \wr C_2$ \( 1 + 3076 T^{2} + 1110214 p^{2} T^{4} + 3076 p^{14} T^{6} + p^{28} T^{8} \)
7$C_2^2 \wr C_2$ \( 1 + 2110260 T^{2} + 2400466154822 T^{4} + 2110260 p^{14} T^{6} + p^{28} T^{8} \)
11$C_2^2 \wr C_2$ \( 1 + 14988652 T^{2} + 812407048031862 T^{4} + 14988652 p^{14} T^{6} + p^{28} T^{8} \)
13$D_{4}$ \( ( 1 + 356 T - 31567218 T^{2} + 356 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 - 19300 T + 756703910 T^{2} - 19300 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
19$C_2^2 \wr C_2$ \( 1 + 2044755436 T^{2} + 2266697792208278166 T^{4} + 2044755436 p^{14} T^{6} + p^{28} T^{8} \)
23$C_2^2 \wr C_2$ \( 1 + 3331906996 T^{2} + 2668149212308037766 T^{4} + 3331906996 p^{14} T^{6} + p^{28} T^{8} \)
29$D_{4}$ \( ( 1 - 161556 T + 40396454158 T^{2} - 161556 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
31$C_2^2 \wr C_2$ \( 1 + 50814705372 T^{2} + \)\(21\!\cdots\!82\)\( T^{4} + 50814705372 p^{14} T^{6} + p^{28} T^{8} \)
37$D_{4}$ \( ( 1 - 465356 T + 243374422206 T^{2} - 465356 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 124628 T - 153459370858 T^{2} + 124628 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
43$C_2^2 \wr C_2$ \( 1 + 757828028260 T^{2} + \)\(28\!\cdots\!42\)\( T^{4} + 757828028260 p^{14} T^{6} + p^{28} T^{8} \)
47$C_2^2 \wr C_2$ \( 1 + 1621179774100 T^{2} + \)\(11\!\cdots\!02\)\( T^{4} + 1621179774100 p^{14} T^{6} + p^{28} T^{8} \)
53$D_{4}$ \( ( 1 - 1047084 T + 1959788175838 T^{2} - 1047084 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
59$C_2^2 \wr C_2$ \( 1 + 3681272468428 T^{2} + \)\(94\!\cdots\!82\)\( T^{4} + 3681272468428 p^{14} T^{6} + p^{28} T^{8} \)
61$D_{4}$ \( ( 1 - 2078572 T + 4108059732942 T^{2} - 2078572 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
67$C_2^2 \wr C_2$ \( 1 + 11129688262980 T^{2} + \)\(85\!\cdots\!62\)\( T^{4} + 11129688262980 p^{14} T^{6} + p^{28} T^{8} \)
71$C_2^2 \wr C_2$ \( 1 - 8265815740868 T^{2} - \)\(10\!\cdots\!78\)\( T^{4} - 8265815740868 p^{14} T^{6} + p^{28} T^{8} \)
73$D_{4}$ \( ( 1 + 5475180 T + 27583238039510 T^{2} + 5475180 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
79$C_2^2 \wr C_2$ \( 1 + 32374299161148 T^{2} + \)\(82\!\cdots\!62\)\( T^{4} + 32374299161148 p^{14} T^{6} + p^{28} T^{8} \)
83$C_2^2 \wr C_2$ \( 1 + 26859892572676 T^{2} + \)\(16\!\cdots\!46\)\( T^{4} + 26859892572676 p^{14} T^{6} + p^{28} T^{8} \)
89$D_{4}$ \( ( 1 + 9599020 T + 89991532392758 T^{2} + 9599020 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 5758708 T + 89575878531942 T^{2} - 5758708 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
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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

−7.36166620341139290153540987930, −6.73732314903333968316709282403, −6.58968529466855142179638881153, −6.37601576934050007875397282117, −6.18178411299927043155762426956, −5.75649491186284742139558158772, −5.61277635062671651404458407403, −5.58502323256029383903358091952, −5.34081118499612346382252591610, −4.71019483226676165119399312338, −4.64312892098270968500196904537, −4.46102710803213720860697040068, −4.08505990979368839083915951217, −3.44295570080219805504224614416, −3.08115567605383214092375173360, −3.05091614583756085408606693853, −3.03606512375591935373806986802, −2.33875859180549431786864585357, −2.12494613212003508826330772352, −2.11378862303786424441850861782, −1.43775193934964235581318468837, −0.952343478417391348949264849810, −0.917004256121880036766401807162, −0.910106348167333850196342117191, −0.03396290328243349205108937178, 0.03396290328243349205108937178, 0.910106348167333850196342117191, 0.917004256121880036766401807162, 0.952343478417391348949264849810, 1.43775193934964235581318468837, 2.11378862303786424441850861782, 2.12494613212003508826330772352, 2.33875859180549431786864585357, 3.03606512375591935373806986802, 3.05091614583756085408606693853, 3.08115567605383214092375173360, 3.44295570080219805504224614416, 4.08505990979368839083915951217, 4.46102710803213720860697040068, 4.64312892098270968500196904537, 4.71019483226676165119399312338, 5.34081118499612346382252591610, 5.58502323256029383903358091952, 5.61277635062671651404458407403, 5.75649491186284742139558158772, 6.18178411299927043155762426956, 6.37601576934050007875397282117, 6.58968529466855142179638881153, 6.73732314903333968316709282403, 7.36166620341139290153540987930

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