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Properties

Label 10-6864e5-1.1-c1e5-0-1
Degree $10$
Conductor $1.524\times 10^{19}$
Sign $1$
Analytic cond. $4.94620\times 10^{8}$
Root an. cond. $7.40333$
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  + 5·3-s + 5-s + 5·7-s + 15·9-s + 5·11-s + 5·13-s + 5·15-s + 4·19-s + 25·21-s + 5·23-s − 10·25-s + 35·27-s + 11·29-s + 8·31-s + 25·33-s + 5·35-s − 8·37-s + 25·39-s − 41-s − 43-s + 15·45-s + 18·47-s + 2·53-s + 5·55-s + 20·57-s + 13·59-s + 9·61-s + ⋯
L(s)  = 1  + 2.88·3-s + 0.447·5-s + 1.88·7-s + 5·9-s + 1.50·11-s + 1.38·13-s + 1.29·15-s + 0.917·19-s + 5.45·21-s + 1.04·23-s − 2·25-s + 6.73·27-s + 2.04·29-s + 1.43·31-s + 4.35·33-s + 0.845·35-s − 1.31·37-s + 4.00·39-s − 0.156·41-s − 0.152·43-s + 2.23·45-s + 2.62·47-s + 0.274·53-s + 0.674·55-s + 2.64·57-s + 1.69·59-s + 1.15·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{5} \cdot 11^{5} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{5} \cdot 11^{5} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{20} \cdot 3^{5} \cdot 11^{5} \cdot 13^{5}\)
Sign: $1$
Analytic conductor: \(4.94620\times 10^{8}\)
Root analytic conductor: \(7.40333\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{20} \cdot 3^{5} \cdot 11^{5} \cdot 13^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(166.6185470\)
\(L(\frac12)\) \(\approx\) \(166.6185470\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{5} \)
11$C_1$ \( ( 1 - T )^{5} \)
13$C_1$ \( ( 1 - T )^{5} \)
good5$C_2 \wr S_5$ \( 1 - T + 11 T^{2} - 8 T^{3} + 62 T^{4} - 34 T^{5} + 62 p T^{6} - 8 p^{2} T^{7} + 11 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 - 5 T + 25 T^{2} - 52 T^{3} + 20 p T^{4} - 174 T^{5} + 20 p^{2} T^{6} - 52 p^{2} T^{7} + 25 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 33 T^{2} - 22 T^{3} + 730 T^{4} - 1004 T^{5} + 730 p T^{6} - 22 p^{2} T^{7} + 33 p^{3} T^{8} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 4 T + 55 T^{2} - 140 T^{3} + 62 p T^{4} - 2400 T^{5} + 62 p^{2} T^{6} - 140 p^{2} T^{7} + 55 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 5 T + 75 T^{2} - 328 T^{3} + 2718 T^{4} - 10342 T^{5} + 2718 p T^{6} - 328 p^{2} T^{7} + 75 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 11 T + 115 T^{2} - 814 T^{3} + 5982 T^{4} - 33602 T^{5} + 5982 p T^{6} - 814 p^{2} T^{7} + 115 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 8 T + 113 T^{2} - 758 T^{3} + 6456 T^{4} - 31364 T^{5} + 6456 p T^{6} - 758 p^{2} T^{7} + 113 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 8 T + 61 T^{2} + 304 T^{3} + 2422 T^{4} + 17744 T^{5} + 2422 p T^{6} + 304 p^{2} T^{7} + 61 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + T + 67 T^{2} + 96 T^{3} + 3632 T^{4} + 1862 T^{5} + 3632 p T^{6} + 96 p^{2} T^{7} + 67 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + T + 19 T^{2} + 54 T^{3} + 844 T^{4} + 2154 T^{5} + 844 p T^{6} + 54 p^{2} T^{7} + 19 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 18 T + 239 T^{2} - 2464 T^{3} + 21406 T^{4} - 154140 T^{5} + 21406 p T^{6} - 2464 p^{2} T^{7} + 239 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 2 T - 15 T^{2} - 808 T^{3} + 90 p T^{4} + 292 p T^{5} + 90 p^{2} T^{6} - 808 p^{2} T^{7} - 15 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 13 T + 323 T^{2} - 2868 T^{3} + 39078 T^{4} - 247982 T^{5} + 39078 p T^{6} - 2868 p^{2} T^{7} + 323 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 9 T + 155 T^{2} - 1356 T^{3} + 16320 T^{4} - 107510 T^{5} + 16320 p T^{6} - 1356 p^{2} T^{7} + 155 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 5 T + 197 T^{2} + 424 T^{3} + 17218 T^{4} + 12174 T^{5} + 17218 p T^{6} + 424 p^{2} T^{7} + 197 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 24 T + 459 T^{2} - 6176 T^{3} + 70978 T^{4} - 638992 T^{5} + 70978 p T^{6} - 6176 p^{2} T^{7} + 459 p^{3} T^{8} - 24 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 13 T + 363 T^{2} + 3716 T^{3} + 52920 T^{4} + 403974 T^{5} + 52920 p T^{6} + 3716 p^{2} T^{7} + 363 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 6 T + 243 T^{2} - 1654 T^{3} + 33406 T^{4} - 170344 T^{5} + 33406 p T^{6} - 1654 p^{2} T^{7} + 243 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 22 T + 335 T^{2} - 3912 T^{3} + 558 p T^{4} - 451364 T^{5} + 558 p^{2} T^{6} - 3912 p^{2} T^{7} + 335 p^{3} T^{8} - 22 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 14 T + 283 T^{2} + 2590 T^{3} + 28848 T^{4} + 233120 T^{5} + 28848 p T^{6} + 2590 p^{2} T^{7} + 283 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 20 T + 593 T^{2} + 7744 T^{3} + 124558 T^{4} + 1125720 T^{5} + 124558 p T^{6} + 7744 p^{2} T^{7} + 593 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.51128744757934121803780445878, −4.39099725110607790853493717177, −4.25351246549694089148415270729, −4.20262397782384183817420275541, −4.17959298896063715097702653429, −3.77186397945876276305305303824, −3.60155283944887728569939878866, −3.52760204514209158790318238752, −3.47904546285437507671587120762, −3.41902660149345206185323283447, −2.98523536701454257071044742742, −2.80851242519473359949302161853, −2.78610223433225189142919449863, −2.68631713333425508368937827088, −2.39906283848335960922679132612, −1.95106675103046680636330258061, −1.93966225224493082027308560893, −1.85117304742605595957822933251, −1.79261957530243163789188962942, −1.76006028555787896386126410809, −1.14364279698951483970563004717, −0.994888079861440283058567961533, −0.968665162519282270619096063035, −0.77377217360794063005062725911, −0.57846996500935879179241950659, 0.57846996500935879179241950659, 0.77377217360794063005062725911, 0.968665162519282270619096063035, 0.994888079861440283058567961533, 1.14364279698951483970563004717, 1.76006028555787896386126410809, 1.79261957530243163789188962942, 1.85117304742605595957822933251, 1.93966225224493082027308560893, 1.95106675103046680636330258061, 2.39906283848335960922679132612, 2.68631713333425508368937827088, 2.78610223433225189142919449863, 2.80851242519473359949302161853, 2.98523536701454257071044742742, 3.41902660149345206185323283447, 3.47904546285437507671587120762, 3.52760204514209158790318238752, 3.60155283944887728569939878866, 3.77186397945876276305305303824, 4.17959298896063715097702653429, 4.20262397782384183817420275541, 4.25351246549694089148415270729, 4.39099725110607790853493717177, 4.51128744757934121803780445878

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