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Properties

Label 10-3344e5-1.1-c1e5-0-1
Degree $10$
Conductor $4.181\times 10^{17}$
Sign $-1$
Analytic cond. $1.35742\times 10^{7}$
Root an. cond. $5.16739$
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  − 3-s − 5·5-s − 6·7-s − 5·9-s − 5·11-s + 4·13-s + 5·15-s − 4·17-s + 5·19-s + 6·21-s − 3·23-s + 3·25-s + 10·27-s + 10·29-s − 11·31-s + 5·33-s + 30·35-s + 37-s − 4·39-s + 2·41-s − 20·43-s + 25·45-s + 20·47-s + 2·49-s + 4·51-s − 14·53-s + 25·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 2.23·5-s − 2.26·7-s − 5/3·9-s − 1.50·11-s + 1.10·13-s + 1.29·15-s − 0.970·17-s + 1.14·19-s + 1.30·21-s − 0.625·23-s + 3/5·25-s + 1.92·27-s + 1.85·29-s − 1.97·31-s + 0.870·33-s + 5.07·35-s + 0.164·37-s − 0.640·39-s + 0.312·41-s − 3.04·43-s + 3.72·45-s + 2.91·47-s + 2/7·49-s + 0.560·51-s − 1.92·53-s + 3.37·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 11^{5} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 11^{5} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{20} \cdot 11^{5} \cdot 19^{5}\)
Sign: $-1$
Analytic conductor: \(1.35742\times 10^{7}\)
Root analytic conductor: \(5.16739\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{20} \cdot 11^{5} \cdot 19^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad2 \( 1 \)
11$C_1$ \( ( 1 + T )^{5} \)
19$C_1$ \( ( 1 - T )^{5} \)
good3$C_2 \wr S_5$ \( 1 + T + 2 p T^{2} + T^{3} + 16 T^{4} - 11 T^{5} + 16 p T^{6} + p^{2} T^{7} + 2 p^{4} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + p T + 22 T^{2} + 67 T^{3} + 196 T^{4} + 439 T^{5} + 196 p T^{6} + 67 p^{2} T^{7} + 22 p^{3} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 6 T + 34 T^{2} + 106 T^{3} + 50 p T^{4} + 832 T^{5} + 50 p^{2} T^{6} + 106 p^{2} T^{7} + 34 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 4 T + 56 T^{2} - 14 p T^{3} + 1376 T^{4} - 3378 T^{5} + 1376 p T^{6} - 14 p^{3} T^{7} + 56 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 4 T + 53 T^{2} + 208 T^{3} + 1562 T^{4} + 4696 T^{5} + 1562 p T^{6} + 208 p^{2} T^{7} + 53 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 3 T + 39 T^{2} - 112 T^{3} - 178 T^{4} - 7542 T^{5} - 178 p T^{6} - 112 p^{2} T^{7} + 39 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 10 T + 108 T^{2} - 504 T^{3} + 4 p^{2} T^{4} - 11922 T^{5} + 4 p^{3} T^{6} - 504 p^{2} T^{7} + 108 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 11 T + 152 T^{2} + 1171 T^{3} + 300 p T^{4} + 52217 T^{5} + 300 p^{2} T^{6} + 1171 p^{2} T^{7} + 152 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - T + 105 T^{2} - 44 T^{3} + 6330 T^{4} - 3606 T^{5} + 6330 p T^{6} - 44 p^{2} T^{7} + 105 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 2 T + 16 T^{2} - 76 T^{3} + 816 T^{4} - 3620 T^{5} + 816 p T^{6} - 76 p^{2} T^{7} + 16 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 20 T + 238 T^{2} + 1800 T^{3} + 11614 T^{4} + 1618 p T^{5} + 11614 p T^{6} + 1800 p^{2} T^{7} + 238 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 20 T + 263 T^{2} - 2672 T^{3} + 23846 T^{4} - 175992 T^{5} + 23846 p T^{6} - 2672 p^{2} T^{7} + 263 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 14 T + 177 T^{2} + 1576 T^{3} + 15906 T^{4} + 118708 T^{5} + 15906 p T^{6} + 1576 p^{2} T^{7} + 177 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 3 T + 131 T^{2} - 200 T^{3} + 5286 T^{4} - 42486 T^{5} + 5286 p T^{6} - 200 p^{2} T^{7} + 131 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 10 T + 281 T^{2} + 1976 T^{3} + 31554 T^{4} + 165916 T^{5} + 31554 p T^{6} + 1976 p^{2} T^{7} + 281 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 9 T + 140 T^{2} + 1585 T^{3} + 16328 T^{4} + 113899 T^{5} + 16328 p T^{6} + 1585 p^{2} T^{7} + 140 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 23 T + 338 T^{2} + 3603 T^{3} + 32304 T^{4} + 260659 T^{5} + 32304 p T^{6} + 3603 p^{2} T^{7} + 338 p^{3} T^{8} + 23 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 25 T^{2} - 16 p T^{3} + 6558 T^{4} - 15136 T^{5} + 6558 p T^{6} - 16 p^{3} T^{7} + 25 p^{3} T^{8} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 44 T + 1143 T^{2} + 20032 T^{3} + 263862 T^{4} + 2652648 T^{5} + 263862 p T^{6} + 20032 p^{2} T^{7} + 1143 p^{3} T^{8} + 44 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 14 T + 346 T^{2} - 3406 T^{3} + 47606 T^{4} - 368596 T^{5} + 47606 p T^{6} - 3406 p^{2} T^{7} + 346 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 27 T + 713 T^{2} + 10780 T^{3} + 152718 T^{4} + 1491426 T^{5} + 152718 p T^{6} + 10780 p^{2} T^{7} + 713 p^{3} T^{8} + 27 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 15 T + 361 T^{2} - 3704 T^{3} + 58310 T^{4} - 473762 T^{5} + 58310 p T^{6} - 3704 p^{2} T^{7} + 361 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.50279502111068171439298385585, −5.40220955246403025801526722449, −5.25158298423880276646228679523, −5.13597671929187469459073753526, −4.77753439191603489589619625981, −4.72349457424629295371279946153, −4.47609912087358215839333661669, −4.26678450715024770039773239756, −4.19752687054131412850554903692, −4.18162946087953513154160259659, −3.71309944073817472794475725733, −3.48937621509181508777761457380, −3.43440064349344977222718886072, −3.38474960759006691610551112364, −3.33450498956722219457749929192, −2.89136108782975314762520037784, −2.81731398990280498041084308796, −2.63982007731356149131265717136, −2.55087113484843366386239488627, −2.50024688272955491319156358870, −1.71300969595905993635703838786, −1.71141064521603140284658458544, −1.48069889285261637294403344002, −1.13242475080857161737610997008, −0.867345782299395195071095357339, 0, 0, 0, 0, 0, 0.867345782299395195071095357339, 1.13242475080857161737610997008, 1.48069889285261637294403344002, 1.71141064521603140284658458544, 1.71300969595905993635703838786, 2.50024688272955491319156358870, 2.55087113484843366386239488627, 2.63982007731356149131265717136, 2.81731398990280498041084308796, 2.89136108782975314762520037784, 3.33450498956722219457749929192, 3.38474960759006691610551112364, 3.43440064349344977222718886072, 3.48937621509181508777761457380, 3.71309944073817472794475725733, 4.18162946087953513154160259659, 4.19752687054131412850554903692, 4.26678450715024770039773239756, 4.47609912087358215839333661669, 4.72349457424629295371279946153, 4.77753439191603489589619625981, 5.13597671929187469459073753526, 5.25158298423880276646228679523, 5.40220955246403025801526722449, 5.50279502111068171439298385585

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