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Properties

Label 10-7605e5-1.1-c1e5-0-1
Degree $10$
Conductor $2.544\times 10^{19}$
Sign $1$
Analytic cond. $8.25812\times 10^{8}$
Root an. cond. $7.79270$
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 + 5·5-s + 7-s − 2·8-s + 10·10-s + 8·11-s + 2·14-s − 2·16-s − 4·19-s + 16·22-s − 6·23-s + 15·25-s + 16·29-s + 9·31-s − 2·32-s + 5·35-s − 4·37-s − 8·38-s − 10·40-s + 6·41-s + 15·43-s − 12·46-s + 10·47-s − 12·49-s + 30·50-s − 20·53-s + 40·55-s + ⋯
L(s)  = 1  + 1.41·2-s + 2.23·5-s + 0.377·7-s − 0.707·8-s + 3.16·10-s + 2.41·11-s + 0.534·14-s − 1/2·16-s − 0.917·19-s + 3.41·22-s − 1.25·23-s + 3·25-s + 2.97·29-s + 1.61·31-s − 0.353·32-s + 0.845·35-s − 0.657·37-s − 1.29·38-s − 1.58·40-s + 0.937·41-s + 2.28·43-s − 1.76·46-s + 1.45·47-s − 1.71·49-s + 4.24·50-s − 2.74·53-s + 5.39·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{5} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{5} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{10} \cdot 5^{5} \cdot 13^{10}\)
Sign: $1$
Analytic conductor: \(8.25812\times 10^{8}\)
Root analytic conductor: \(7.79270\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 3^{10} \cdot 5^{5} \cdot 13^{10} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(63.72134121\)
\(L(\frac12)\) \(\approx\) \(63.72134121\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
5$C_1$ \( ( 1 - T )^{5} \)
13 \( 1 \)
good2$C_2 \wr S_5$ \( 1 - p T + p^{2} T^{2} - 3 p T^{3} + 5 p T^{4} - 7 p T^{5} + 5 p^{2} T^{6} - 3 p^{3} T^{7} + p^{5} T^{8} - p^{5} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 - T + 13 T^{2} - 6 T^{3} + 75 T^{4} - 5 p T^{5} + 75 p T^{6} - 6 p^{2} T^{7} + 13 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 8 T + 5 p T^{2} - 288 T^{3} + 1258 T^{4} - 4424 T^{5} + 1258 p T^{6} - 288 p^{2} T^{7} + 5 p^{4} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 35 T^{2} - 14 T^{3} + 910 T^{4} + 166 T^{5} + 910 p T^{6} - 14 p^{2} T^{7} + 35 p^{3} T^{8} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 4 T + 87 T^{2} + 268 T^{3} + 3150 T^{4} + 7308 T^{5} + 3150 p T^{6} + 268 p^{2} T^{7} + 87 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 6 T + 95 T^{2} + 516 T^{3} + 4030 T^{4} + 17316 T^{5} + 4030 p T^{6} + 516 p^{2} T^{7} + 95 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 16 T + 7 p T^{2} - 1764 T^{3} + 13072 T^{4} - 75142 T^{5} + 13072 p T^{6} - 1764 p^{2} T^{7} + 7 p^{4} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 9 T + 125 T^{2} - 938 T^{3} + 7513 T^{4} - 40255 T^{5} + 7513 p T^{6} - 938 p^{2} T^{7} + 125 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 4 T + 121 T^{2} + 648 T^{3} + 6826 T^{4} + 36856 T^{5} + 6826 p T^{6} + 648 p^{2} T^{7} + 121 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 6 T + 107 T^{2} - 420 T^{3} + 6892 T^{4} - 26166 T^{5} + 6892 p T^{6} - 420 p^{2} T^{7} + 107 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 15 T + 245 T^{2} - 2122 T^{3} + 20133 T^{4} - 124963 T^{5} + 20133 p T^{6} - 2122 p^{2} T^{7} + 245 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 10 T + 145 T^{2} - 24 p T^{3} + 10018 T^{4} - 69286 T^{5} + 10018 p T^{6} - 24 p^{3} T^{7} + 145 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 20 T + 5 p T^{2} + 2368 T^{3} + 17722 T^{4} + 123096 T^{5} + 17722 p T^{6} + 2368 p^{2} T^{7} + 5 p^{4} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 12 T + 253 T^{2} - 2436 T^{3} + 27700 T^{4} - 206658 T^{5} + 27700 p T^{6} - 2436 p^{2} T^{7} + 253 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 11 T + 291 T^{2} - 2278 T^{3} + 34301 T^{4} - 199077 T^{5} + 34301 p T^{6} - 2278 p^{2} T^{7} + 291 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 5 T + 291 T^{2} - 1336 T^{3} + 36125 T^{4} - 134151 T^{5} + 36125 p T^{6} - 1336 p^{2} T^{7} + 291 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 10 T + 333 T^{2} - 2486 T^{3} + 45208 T^{4} - 252030 T^{5} + 45208 p T^{6} - 2486 p^{2} T^{7} + 333 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - T + 129 T^{2} - 148 T^{3} + 13711 T^{4} - 1257 T^{5} + 13711 p T^{6} - 148 p^{2} T^{7} + 129 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 17 T + 369 T^{2} + 4286 T^{3} + 54349 T^{4} + 467343 T^{5} + 54349 p T^{6} + 4286 p^{2} T^{7} + 369 p^{3} T^{8} + 17 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 16 T + 299 T^{2} - 3252 T^{3} + 38638 T^{4} - 325648 T^{5} + 38638 p T^{6} - 3252 p^{2} T^{7} + 299 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 4 T + 343 T^{2} - 1346 T^{3} + 52048 T^{4} - 176274 T^{5} + 52048 p T^{6} - 1346 p^{2} T^{7} + 343 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 11 T + 355 T^{2} + 2162 T^{3} + 48109 T^{4} + 204751 T^{5} + 48109 p T^{6} + 2162 p^{2} T^{7} + 355 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.49331261256861775289295550525, −4.46928638090898614776760493647, −4.29226938898779484166376043229, −4.28631626119401157245744777129, −4.24281926830327160547336005357, −3.88795853516402148307989725995, −3.64858200525635880831593468605, −3.62831646652357033782172202897, −3.61476376959821422347387920779, −3.20375829656952856789392041089, −3.05681849846451876017184574493, −2.79249095885892424265788206938, −2.66913865729436401756597192896, −2.48186828730974740579568751810, −2.40873308154956628981895829592, −2.36951207230923236856669584712, −1.91878761188370491034481789321, −1.71658231911794869998818737466, −1.71640980442197004778416454081, −1.40634333539650549797503295885, −1.37364747249870038205659075213, −0.907027760340747847996422980849, −0.815166266494946362532299838306, −0.62345007572729809692428891826, −0.43670704102471623582003862819, 0.43670704102471623582003862819, 0.62345007572729809692428891826, 0.815166266494946362532299838306, 0.907027760340747847996422980849, 1.37364747249870038205659075213, 1.40634333539650549797503295885, 1.71640980442197004778416454081, 1.71658231911794869998818737466, 1.91878761188370491034481789321, 2.36951207230923236856669584712, 2.40873308154956628981895829592, 2.48186828730974740579568751810, 2.66913865729436401756597192896, 2.79249095885892424265788206938, 3.05681849846451876017184574493, 3.20375829656952856789392041089, 3.61476376959821422347387920779, 3.62831646652357033782172202897, 3.64858200525635880831593468605, 3.88795853516402148307989725995, 4.24281926830327160547336005357, 4.28631626119401157245744777129, 4.29226938898779484166376043229, 4.46928638090898614776760493647, 4.49331261256861775289295550525

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