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Properties

Label 10-8280e5-1.1-c1e5-0-1
Degree $10$
Conductor $3.892\times 10^{19}$
Sign $1$
Analytic cond. $1.26338\times 10^{9}$
Root an. cond. $8.13118$
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  + 5·5-s − 2·7-s + 11-s + 4·13-s − 4·17-s + 7·19-s + 5·23-s + 15·25-s − 4·29-s + 19·31-s − 10·35-s + 15·37-s − 25·41-s + 11·47-s − 3·49-s − 3·53-s + 5·55-s + 59-s − 5·61-s + 20·65-s + 9·67-s − 71-s − 73-s − 2·77-s − 2·79-s + 45·83-s − 20·85-s + ⋯
L(s)  = 1  + 2.23·5-s − 0.755·7-s + 0.301·11-s + 1.10·13-s − 0.970·17-s + 1.60·19-s + 1.04·23-s + 3·25-s − 0.742·29-s + 3.41·31-s − 1.69·35-s + 2.46·37-s − 3.90·41-s + 1.60·47-s − 3/7·49-s − 0.412·53-s + 0.674·55-s + 0.130·59-s − 0.640·61-s + 2.48·65-s + 1.09·67-s − 0.118·71-s − 0.117·73-s − 0.227·77-s − 0.225·79-s + 4.93·83-s − 2.16·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{10} \cdot 5^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{10} \cdot 5^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{15} \cdot 3^{10} \cdot 5^{5} \cdot 23^{5}\)
Sign: $1$
Analytic conductor: \(1.26338\times 10^{9}\)
Root analytic conductor: \(8.13118\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{15} \cdot 3^{10} \cdot 5^{5} \cdot 23^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(25.51183853\)
\(L(\frac12)\) \(\approx\) \(25.51183853\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{5} \)
23$C_1$ \( ( 1 - T )^{5} \)
good7$C_2 \wr S_5$ \( 1 + 2 T + p T^{2} - T^{3} + 30 T^{4} + 46 T^{5} + 30 p T^{6} - p^{2} T^{7} + p^{4} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - T + 20 T^{2} - 16 T^{3} + 227 T^{4} - 174 T^{5} + 227 p T^{6} - 16 p^{2} T^{7} + 20 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 4 T + 19 T^{2} - 133 T^{3} + 496 T^{4} - 1606 T^{5} + 496 p T^{6} - 133 p^{2} T^{7} + 19 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 4 T + 39 T^{2} + 115 T^{3} + 406 T^{4} + 1566 T^{5} + 406 p T^{6} + 115 p^{2} T^{7} + 39 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 7 T + 54 T^{2} - 352 T^{3} + 1949 T^{4} - 7810 T^{5} + 1949 p T^{6} - 352 p^{2} T^{7} + 54 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 4 T + 104 T^{2} + 500 T^{3} + 4907 T^{4} + 22264 T^{5} + 4907 p T^{6} + 500 p^{2} T^{7} + 104 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 19 T + 227 T^{2} - 2173 T^{3} + 16253 T^{4} - 98336 T^{5} + 16253 p T^{6} - 2173 p^{2} T^{7} + 227 p^{3} T^{8} - 19 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 15 T + 249 T^{2} - 2256 T^{3} + 20618 T^{4} - 125810 T^{5} + 20618 p T^{6} - 2256 p^{2} T^{7} + 249 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 25 T + 417 T^{2} + 4753 T^{3} + 42913 T^{4} + 303514 T^{5} + 42913 p T^{6} + 4753 p^{2} T^{7} + 417 p^{3} T^{8} + 25 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2$ \( ( 1 + p T^{2} )^{5} \)
47$C_2 \wr S_5$ \( 1 - 11 T + 125 T^{2} - 952 T^{3} + 5780 T^{4} - 41402 T^{5} + 5780 p T^{6} - 952 p^{2} T^{7} + 125 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 3 T + 105 T^{2} + 196 T^{3} + 8458 T^{4} + 24194 T^{5} + 8458 p T^{6} + 196 p^{2} T^{7} + 105 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - T + 153 T^{2} + 14236 T^{4} - 6606 T^{5} + 14236 p T^{6} + 153 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 5 T + 190 T^{2} + 652 T^{3} + 18257 T^{4} + 49998 T^{5} + 18257 p T^{6} + 652 p^{2} T^{7} + 190 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 9 T + 209 T^{2} - 1980 T^{3} + 24396 T^{4} - 176326 T^{5} + 24396 p T^{6} - 1980 p^{2} T^{7} + 209 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + T + 141 T^{2} - 123 T^{3} + 12967 T^{4} - 23580 T^{5} + 12967 p T^{6} - 123 p^{2} T^{7} + 141 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + T + 207 T^{2} + 20 T^{3} + 20760 T^{4} - 9066 T^{5} + 20760 p T^{6} + 20 p^{2} T^{7} + 207 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 2 T + 267 T^{2} + 760 T^{3} + 34506 T^{4} + 94092 T^{5} + 34506 p T^{6} + 760 p^{2} T^{7} + 267 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 45 T + 1199 T^{2} - 21528 T^{3} + 290714 T^{4} - 2994854 T^{5} + 290714 p T^{6} - 21528 p^{2} T^{7} + 1199 p^{3} T^{8} - 45 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 6 T + 309 T^{2} + 1624 T^{3} + 45458 T^{4} + 202212 T^{5} + 45458 p T^{6} + 1624 p^{2} T^{7} + 309 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 25 T + 446 T^{2} - 5536 T^{3} + 65833 T^{4} - 653150 T^{5} + 65833 p T^{6} - 5536 p^{2} T^{7} + 446 p^{3} T^{8} - 25 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.60844251710145976019409731613, −4.45637650079522595400150967894, −4.27106437084947844768804702335, −4.17590576194721896846719781349, −4.09398215309837419589631421432, −3.58958670578191695976521264071, −3.46270635553451443488052689074, −3.45060982657215529869078400413, −3.38109427701902601368972664823, −3.31359731956540054699556897469, −2.86847756713323311857326493321, −2.83240130784197051410394809298, −2.55790502968397206883568069156, −2.55337992884593123279117706974, −2.46755196347086650262562836273, −2.06388958121212960585122426437, −1.92278472834604443669717788447, −1.68837990145725578811305610802, −1.61564442318116889645919096568, −1.56441827587576104151232641959, −0.990846893053214022973120212946, −0.870580025507061478568019292232, −0.841527663918498789589108559110, −0.68406165400172018254651081345, −0.33918727466675367037138430248, 0.33918727466675367037138430248, 0.68406165400172018254651081345, 0.841527663918498789589108559110, 0.870580025507061478568019292232, 0.990846893053214022973120212946, 1.56441827587576104151232641959, 1.61564442318116889645919096568, 1.68837990145725578811305610802, 1.92278472834604443669717788447, 2.06388958121212960585122426437, 2.46755196347086650262562836273, 2.55337992884593123279117706974, 2.55790502968397206883568069156, 2.83240130784197051410394809298, 2.86847756713323311857326493321, 3.31359731956540054699556897469, 3.38109427701902601368972664823, 3.45060982657215529869078400413, 3.46270635553451443488052689074, 3.58958670578191695976521264071, 4.09398215309837419589631421432, 4.17590576194721896846719781349, 4.27106437084947844768804702335, 4.45637650079522595400150967894, 4.60844251710145976019409731613

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