Properties

Label 10-5225e5-1.1-c1e5-0-0
Degree $10$
Conductor $3.894\times 10^{18}$
Sign $1$
Analytic cond. $1.26420\times 10^{8}$
Root an. cond. $6.45924$
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 − 3-s + 2·6-s − 6·7-s + 2·8-s − 5·9-s + 5·11-s − 4·13-s + 12·14-s − 16-s + 4·17-s + 10·18-s − 5·19-s + 6·21-s − 10·22-s − 3·23-s − 2·24-s + 8·26-s + 10·27-s + 10·29-s + 11·31-s − 5·33-s − 8·34-s − 37-s + 10·38-s + 4·39-s + 2·41-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 0.816·6-s − 2.26·7-s + 0.707·8-s − 5/3·9-s + 1.50·11-s − 1.10·13-s + 3.20·14-s − 1/4·16-s + 0.970·17-s + 2.35·18-s − 1.14·19-s + 1.30·21-s − 2.13·22-s − 0.625·23-s − 0.408·24-s + 1.56·26-s + 1.92·27-s + 1.85·29-s + 1.97·31-s − 0.870·33-s − 1.37·34-s − 0.164·37-s + 1.62·38-s + 0.640·39-s + 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{10} \cdot 11^{5} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{10} \cdot 11^{5} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(5^{10} \cdot 11^{5} \cdot 19^{5}\)
Sign: $1$
Analytic conductor: \(1.26420\times 10^{8}\)
Root analytic conductor: \(6.45924\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 5^{10} \cdot 11^{5} \cdot 19^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.147391030\)
\(L(\frac12)\) \(\approx\) \(1.147391030\)
\(L(\frac{3}{2})\) 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)$
bad5 \( 1 \)
11$C_1$ \( ( 1 - T )^{5} \)
19$C_1$ \( ( 1 + T )^{5} \)
good2$C_2 \wr S_5$ \( 1 + p T + p^{2} T^{2} + 3 p T^{3} + 9 T^{4} + 3 p^{2} T^{5} + 9 p T^{6} + 3 p^{3} T^{7} + p^{5} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
3$C_2 \wr S_5$ \( 1 + T + 2 p T^{2} + T^{3} + 16 T^{4} - 11 T^{5} + 16 p T^{6} + p^{2} T^{7} + 2 p^{4} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
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

−4.79967915810279737149034207567, −4.62334797430252589198334033371, −4.44565312023786055332977118030, −4.38786337033254034465079351574, −4.33386948081024750066131602125, −4.13661201601547363016097718451, −3.78314796132251823219006366781, −3.49830225933913064279879161242, −3.49719443742333774873125336578, −3.36342436189195674930543547649, −3.17194334125720527362686265134, −3.02935104356661372520351943653, −2.95851566963601609146117106503, −2.64365953832161833016579236031, −2.36301497932260393340148642980, −2.33284885720576716628473793284, −2.23008775285929655213006175511, −1.74588089950340618641605379373, −1.67715722736293853044114636910, −1.38269143537039590031412517567, −0.972954146292420588465780984368, −0.63621540667423313246754879428, −0.52060095119361052863498751755, −0.46053552493497528500822201611, −0.45811823955804093937271218957, 0.45811823955804093937271218957, 0.46053552493497528500822201611, 0.52060095119361052863498751755, 0.63621540667423313246754879428, 0.972954146292420588465780984368, 1.38269143537039590031412517567, 1.67715722736293853044114636910, 1.74588089950340618641605379373, 2.23008775285929655213006175511, 2.33284885720576716628473793284, 2.36301497932260393340148642980, 2.64365953832161833016579236031, 2.95851566963601609146117106503, 3.02935104356661372520351943653, 3.17194334125720527362686265134, 3.36342436189195674930543547649, 3.49719443742333774873125336578, 3.49830225933913064279879161242, 3.78314796132251823219006366781, 4.13661201601547363016097718451, 4.33386948081024750066131602125, 4.38786337033254034465079351574, 4.44565312023786055332977118030, 4.62334797430252589198334033371, 4.79967915810279737149034207567

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