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Properties

Label 10-7728e5-1.1-c1e5-0-0
Degree $10$
Conductor $2.756\times 10^{19}$
Sign $1$
Analytic cond. $8.94789\times 10^{8}$
Root an. cond. $7.85546$
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  − 5·3-s − 6·5-s − 5·7-s + 15·9-s + 4·13-s + 30·15-s + 8·17-s − 10·19-s + 25·21-s + 5·23-s + 11·25-s − 35·27-s − 3·29-s − 2·31-s + 30·35-s + 3·37-s − 20·39-s + 3·41-s − 24·43-s − 90·45-s + 5·47-s + 15·49-s − 40·51-s + 9·53-s + 50·57-s − 3·59-s + 61-s + ⋯
L(s)  = 1  − 2.88·3-s − 2.68·5-s − 1.88·7-s + 5·9-s + 1.10·13-s + 7.74·15-s + 1.94·17-s − 2.29·19-s + 5.45·21-s + 1.04·23-s + 11/5·25-s − 6.73·27-s − 0.557·29-s − 0.359·31-s + 5.07·35-s + 0.493·37-s − 3.20·39-s + 0.468·41-s − 3.65·43-s − 13.4·45-s + 0.729·47-s + 15/7·49-s − 5.60·51-s + 1.23·53-s + 6.62·57-s − 0.390·59-s + 0.128·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{5} \cdot 7^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{5} \cdot 7^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{20} \cdot 3^{5} \cdot 7^{5} \cdot 23^{5}\)
Sign: $1$
Analytic conductor: \(8.94789\times 10^{8}\)
Root analytic conductor: \(7.85546\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{20} \cdot 3^{5} \cdot 7^{5} \cdot 23^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1318251711\)
\(L(\frac12)\) \(\approx\) \(0.1318251711\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 + T )^{5} \)
7$C_1$ \( ( 1 + T )^{5} \)
23$C_1$ \( ( 1 - T )^{5} \)
good5$C_2 \wr S_5$ \( 1 + 6 T + p^{2} T^{2} + 77 T^{3} + 206 T^{4} + 474 T^{5} + 206 p T^{6} + 77 p^{2} T^{7} + p^{5} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 6 T^{2} + 16 T^{3} + 29 T^{4} - 672 T^{5} + 29 p T^{6} + 16 p^{2} T^{7} - 6 p^{3} T^{8} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 4 T + 25 T^{2} - 113 T^{3} + 562 T^{4} - 1870 T^{5} + 562 p T^{6} - 113 p^{2} T^{7} + 25 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 8 T + 60 T^{2} - 304 T^{3} + 1635 T^{4} - 6736 T^{5} + 1635 p T^{6} - 304 p^{2} T^{7} + 60 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 10 T + 64 T^{2} + 348 T^{3} + 2035 T^{4} + 10100 T^{5} + 2035 p T^{6} + 348 p^{2} T^{7} + 64 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 3 T + 44 T^{2} - 15 T^{3} + 2135 T^{4} + 4280 T^{5} + 2135 p T^{6} - 15 p^{2} T^{7} + 44 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 2 T + 106 T^{2} + 226 T^{3} + 5557 T^{4} + 9848 T^{5} + 5557 p T^{6} + 226 p^{2} T^{7} + 106 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 3 T + 84 T^{2} - 81 T^{3} + 4991 T^{4} - 7976 T^{5} + 4991 p T^{6} - 81 p^{2} T^{7} + 84 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 3 T + 104 T^{2} - 129 T^{3} + 6899 T^{4} - 10688 T^{5} + 6899 p T^{6} - 129 p^{2} T^{7} + 104 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 24 T + 355 T^{2} + 3815 T^{3} + 33046 T^{4} + 235434 T^{5} + 33046 p T^{6} + 3815 p^{2} T^{7} + 355 p^{3} T^{8} + 24 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 5 T + 113 T^{2} - 192 T^{3} + 4008 T^{4} + 4298 T^{5} + 4008 p T^{6} - 192 p^{2} T^{7} + 113 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 9 T + 80 T^{2} - 408 T^{3} + 7007 T^{4} - 56254 T^{5} + 7007 p T^{6} - 408 p^{2} T^{7} + 80 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 3 T + 136 T^{2} + 458 T^{3} + 10091 T^{4} + 35814 T^{5} + 10091 p T^{6} + 458 p^{2} T^{7} + 136 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - T + 204 T^{2} - 210 T^{3} + 20939 T^{4} - 17850 T^{5} + 20939 p T^{6} - 210 p^{2} T^{7} + 204 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 9 T + 210 T^{2} + 920 T^{3} + 15157 T^{4} + 38126 T^{5} + 15157 p T^{6} + 920 p^{2} T^{7} + 210 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + T + 174 T^{2} + 274 T^{3} + 17857 T^{4} + 30874 T^{5} + 17857 p T^{6} + 274 p^{2} T^{7} + 174 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 2 T + 218 T^{2} - 808 T^{3} + 24909 T^{4} - 85820 T^{5} + 24909 p T^{6} - 808 p^{2} T^{7} + 218 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 334 T^{2} - 16 T^{3} + 48785 T^{4} - 1504 T^{5} + 48785 p T^{6} - 16 p^{2} T^{7} + 334 p^{3} T^{8} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 34 T + 732 T^{2} + 11384 T^{3} + 140223 T^{4} + 1409900 T^{5} + 140223 p T^{6} + 11384 p^{2} T^{7} + 732 p^{3} T^{8} + 34 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 11 T + 260 T^{2} - 1214 T^{3} + 24683 T^{4} - 66222 T^{5} + 24683 p T^{6} - 1214 p^{2} T^{7} + 260 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 47 T + 1172 T^{2} - 20549 T^{3} + 282435 T^{4} - 3106952 T^{5} + 282435 p T^{6} - 20549 p^{2} T^{7} + 1172 p^{3} T^{8} - 47 p^{4} T^{9} + p^{5} T^{10} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.52492767741483929568266531894, −4.35492865404284588760414426758, −4.30642653584781865656582889669, −4.28292997078011104311109381997, −4.04615392213019867465680190949, −3.73641375688454048038509341030, −3.72676501159648313249508079585, −3.64983009170717565517169462223, −3.54090092107996657745162497137, −3.41451130176187605649505031326, −3.09469632353544560134861995370, −2.99346176723810677507822285932, −2.78075713017077747452494213559, −2.53884512894217958069783722349, −2.50377414338157838836436987277, −1.98686803841350165897613031885, −1.70794587150615257709954379124, −1.70244538364377588230630044803, −1.54695494321202444580371487863, −1.27303659523327064095789331848, −0.809291395394980103022870585501, −0.77122110585245585765335016916, −0.59457269870187268847450115063, −0.23168702351168136629563322453, −0.17797704720568139974971665194, 0.17797704720568139974971665194, 0.23168702351168136629563322453, 0.59457269870187268847450115063, 0.77122110585245585765335016916, 0.809291395394980103022870585501, 1.27303659523327064095789331848, 1.54695494321202444580371487863, 1.70244538364377588230630044803, 1.70794587150615257709954379124, 1.98686803841350165897613031885, 2.50377414338157838836436987277, 2.53884512894217958069783722349, 2.78075713017077747452494213559, 2.99346176723810677507822285932, 3.09469632353544560134861995370, 3.41451130176187605649505031326, 3.54090092107996657745162497137, 3.64983009170717565517169462223, 3.72676501159648313249508079585, 3.73641375688454048038509341030, 4.04615392213019867465680190949, 4.28292997078011104311109381997, 4.30642653584781865656582889669, 4.35492865404284588760414426758, 4.52492767741483929568266531894

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