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Properties

Label 10-6045e5-1.1-c1e5-0-0
Degree $10$
Conductor $8.072\times 10^{18}$
Sign $-1$
Analytic cond. $2.62039\times 10^{8}$
Root an. cond. $6.94763$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $5$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·3-s − 2·4-s + 5·5-s − 7-s + 2·8-s + 15·9-s − 9·11-s − 10·12-s − 5·13-s + 25·15-s − 2·16-s − 17·17-s − 2·19-s − 10·20-s − 5·21-s − 17·23-s + 10·24-s + 15·25-s + 35·27-s + 2·28-s − 6·29-s − 5·31-s − 8·32-s − 45·33-s − 5·35-s − 30·36-s − 3·37-s + ⋯
L(s)  = 1  + 2.88·3-s − 4-s + 2.23·5-s − 0.377·7-s + 0.707·8-s + 5·9-s − 2.71·11-s − 2.88·12-s − 1.38·13-s + 6.45·15-s − 1/2·16-s − 4.12·17-s − 0.458·19-s − 2.23·20-s − 1.09·21-s − 3.54·23-s + 2.04·24-s + 3·25-s + 6.73·27-s + 0.377·28-s − 1.11·29-s − 0.898·31-s − 1.41·32-s − 7.83·33-s − 0.845·35-s − 5·36-s − 0.493·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 5^{5} \cdot 13^{5} \cdot 31^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 5^{5} \cdot 13^{5} \cdot 31^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{5} \cdot 5^{5} \cdot 13^{5} \cdot 31^{5}\)
Sign: $-1$
Analytic conductor: \(2.62039\times 10^{8}\)
Root analytic conductor: \(6.94763\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 3^{5} \cdot 5^{5} \cdot 13^{5} \cdot 31^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{5} \)
5$C_1$ \( ( 1 - T )^{5} \)
13$C_1$ \( ( 1 + T )^{5} \)
31$C_1$ \( ( 1 + T )^{5} \)
good2$C_2 \wr S_5$ \( 1 + p T^{2} - p T^{3} + 3 p T^{4} + 3 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + T + 3 T^{2} - 6 T^{3} + 59 T^{4} + 129 T^{5} + 59 p T^{6} - 6 p^{2} T^{7} + 3 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 9 T + 67 T^{2} + 348 T^{3} + 1542 T^{4} + 5462 T^{5} + 1542 p T^{6} + 348 p^{2} T^{7} + 67 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + p T + 163 T^{2} + 1036 T^{3} + 5196 T^{4} + 22286 T^{5} + 5196 p T^{6} + 1036 p^{2} T^{7} + 163 p^{3} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 2 T + 59 T^{2} + 142 T^{3} + 1804 T^{4} + 3708 T^{5} + 1804 p T^{6} + 142 p^{2} T^{7} + 59 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 17 T + 163 T^{2} + 1108 T^{3} + 6330 T^{4} + 31910 T^{5} + 6330 p T^{6} + 1108 p^{2} T^{7} + 163 p^{3} T^{8} + 17 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 6 T + 5 p T^{2} + 648 T^{3} + 8349 T^{4} + 27500 T^{5} + 8349 p T^{6} + 648 p^{2} T^{7} + 5 p^{4} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 3 T + 75 T^{2} - 56 T^{3} + 2760 T^{4} - 6446 T^{5} + 2760 p T^{6} - 56 p^{2} T^{7} + 75 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 9 T + 191 T^{2} - 1332 T^{3} + 14963 T^{4} - 79027 T^{5} + 14963 p T^{6} - 1332 p^{2} T^{7} + 191 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 8 T + 75 T^{2} + 176 T^{3} - 2751 T^{4} + 41394 T^{5} - 2751 p T^{6} + 176 p^{2} T^{7} + 75 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 2 T - 69 T^{2} + 330 T^{3} + 2060 T^{4} - 27560 T^{5} + 2060 p T^{6} + 330 p^{2} T^{7} - 69 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 51 T + 1265 T^{2} + 20042 T^{3} + 225450 T^{4} + 1888428 T^{5} + 225450 p T^{6} + 20042 p^{2} T^{7} + 1265 p^{3} T^{8} + 51 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 6 T + 181 T^{2} + 564 T^{3} + 13773 T^{4} + 26912 T^{5} + 13773 p T^{6} + 564 p^{2} T^{7} + 181 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 23 T + 475 T^{2} + 6034 T^{3} + 68444 T^{4} + 564864 T^{5} + 68444 p T^{6} + 6034 p^{2} T^{7} + 475 p^{3} T^{8} + 23 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 185 T^{2} - 850 T^{3} + 14129 T^{4} - 111424 T^{5} + 14129 p T^{6} - 850 p^{2} T^{7} + 185 p^{3} T^{8} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 9 T + 187 T^{2} - 1500 T^{3} + 22434 T^{4} - 143254 T^{5} + 22434 p T^{6} - 1500 p^{2} T^{7} + 187 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 16 T + 223 T^{2} + 1774 T^{3} + 21430 T^{4} + 168592 T^{5} + 21430 p T^{6} + 1774 p^{2} T^{7} + 223 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 21 T + 453 T^{2} - 5338 T^{3} + 66352 T^{4} - 562528 T^{5} + 66352 p T^{6} - 5338 p^{2} T^{7} + 453 p^{3} T^{8} - 21 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 44 T + 1075 T^{2} + 18102 T^{3} + 231457 T^{4} + 2351242 T^{5} + 231457 p T^{6} + 18102 p^{2} T^{7} + 1075 p^{3} T^{8} + 44 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 65 T + 2071 T^{2} + 42328 T^{3} + 615832 T^{4} + 6681046 T^{5} + 615832 p T^{6} + 42328 p^{2} T^{7} + 2071 p^{3} T^{8} + 65 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 11 T + 227 T^{2} - 1916 T^{3} + 21839 T^{4} - 186229 T^{5} + 21839 p T^{6} - 1916 p^{2} T^{7} + 227 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.93512678663117921663179368455, −4.76692730262159783646878405843, −4.73357711413013048565129477522, −4.64405343751532345762835401915, −4.62612484257603768333307848895, −4.24267938061356048817164598937, −4.11480350066072214274626526196, −4.08049291419120175787386968306, −4.04728195345881924041602708595, −4.01692592685457982513145744947, −3.26052861260940889243993866368, −3.19733672142511186619042347548, −3.17938573734378262172768217219, −3.00953093497330524341368701081, −2.72879249005293419365055704242, −2.64098738662822787086599050901, −2.57013546153130778547545625346, −2.22291265811163532059722729414, −2.17295093277087711573649836314, −2.10313396306198347686131927498, −1.86192440374516445508171972916, −1.72164198999494644244538349344, −1.64486726034568626005775204858, −1.35325084385881051905563468079, −1.23860460411415196913081887870, 0, 0, 0, 0, 0, 1.23860460411415196913081887870, 1.35325084385881051905563468079, 1.64486726034568626005775204858, 1.72164198999494644244538349344, 1.86192440374516445508171972916, 2.10313396306198347686131927498, 2.17295093277087711573649836314, 2.22291265811163532059722729414, 2.57013546153130778547545625346, 2.64098738662822787086599050901, 2.72879249005293419365055704242, 3.00953093497330524341368701081, 3.17938573734378262172768217219, 3.19733672142511186619042347548, 3.26052861260940889243993866368, 4.01692592685457982513145744947, 4.04728195345881924041602708595, 4.08049291419120175787386968306, 4.11480350066072214274626526196, 4.24267938061356048817164598937, 4.62612484257603768333307848895, 4.64405343751532345762835401915, 4.73357711413013048565129477522, 4.76692730262159783646878405843, 4.93512678663117921663179368455

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