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Properties

Label 10-1143e5-1.1-c1e5-0-1
Degree $10$
Conductor $1.951\times 10^{15}$
Sign $-1$
Analytic cond. $63331.0$
Root an. cond. $3.02107$
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  − 2·2-s − 5-s + 2·8-s + 2·10-s − 14·11-s − 5·13-s − 16-s − 4·17-s + 4·19-s + 28·22-s − 15·23-s − 15·25-s + 10·26-s − 9·29-s + 3·31-s + 8·34-s − 5·37-s − 8·38-s − 2·40-s − 4·41-s + 10·43-s + 30·46-s + 4·47-s − 22·49-s + 30·50-s − 3·53-s + 14·55-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.447·5-s + 0.707·8-s + 0.632·10-s − 4.22·11-s − 1.38·13-s − 1/4·16-s − 0.970·17-s + 0.917·19-s + 5.96·22-s − 3.12·23-s − 3·25-s + 1.96·26-s − 1.67·29-s + 0.538·31-s + 1.37·34-s − 0.821·37-s − 1.29·38-s − 0.316·40-s − 0.624·41-s + 1.52·43-s + 4.42·46-s + 0.583·47-s − 3.14·49-s + 4.24·50-s − 0.412·53-s + 1.88·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 127^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 127^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{10} \cdot 127^{5}\)
Sign: $-1$
Analytic conductor: \(63331.0\)
Root analytic conductor: \(3.02107\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 3^{10} \cdot 127^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
127$C_1$ \( ( 1 - T )^{5} \)
good2$C_2 \wr S_5$ \( 1 + p T + p^{2} T^{2} + 3 p T^{3} + 9 T^{4} + 3 p^{2} T^{5} + 9 p T^{6} + 3 p^{3} T^{7} + p^{5} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + T + 16 T^{2} + 9 T^{3} + 122 T^{4} + 41 T^{5} + 122 p T^{6} + 9 p^{2} T^{7} + 16 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 22 T^{2} - 4 T^{3} + 250 T^{4} - 54 T^{5} + 250 p T^{6} - 4 p^{2} T^{7} + 22 p^{3} T^{8} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 14 T + 118 T^{2} + 712 T^{3} + 302 p T^{4} + 12268 T^{5} + 302 p^{2} T^{6} + 712 p^{2} T^{7} + 118 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 5 T + 62 T^{2} + 227 T^{3} + 1564 T^{4} + 4231 T^{5} + 1564 p T^{6} + 227 p^{2} T^{7} + 62 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 4 T + 53 T^{2} + 208 T^{3} + 1562 T^{4} + 4696 T^{5} + 1562 p T^{6} + 208 p^{2} T^{7} + 53 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 4 T + 55 T^{2} - 240 T^{3} + 1522 T^{4} - 6488 T^{5} + 1522 p T^{6} - 240 p^{2} T^{7} + 55 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 15 T + 179 T^{2} + 1384 T^{3} + 9258 T^{4} + 47202 T^{5} + 9258 p T^{6} + 1384 p^{2} T^{7} + 179 p^{3} T^{8} + 15 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 9 T + 114 T^{2} + 697 T^{3} + 198 p T^{4} + 27567 T^{5} + 198 p^{2} T^{6} + 697 p^{2} T^{7} + 114 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 3 T + 55 T^{2} - 376 T^{3} + 1734 T^{4} - 15706 T^{5} + 1734 p T^{6} - 376 p^{2} T^{7} + 55 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 5 T + 94 T^{2} + 87 T^{3} + 2920 T^{4} - 6345 T^{5} + 2920 p T^{6} + 87 p^{2} T^{7} + 94 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 4 T + 89 T^{2} + 336 T^{3} + 3966 T^{4} + 16472 T^{5} + 3966 p T^{6} + 336 p^{2} T^{7} + 89 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 10 T + 126 T^{2} - 974 T^{3} + 8670 T^{4} - 58056 T^{5} + 8670 p T^{6} - 974 p^{2} T^{7} + 126 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 4 T + 186 T^{2} - 638 T^{3} + 15762 T^{4} - 42452 T^{5} + 15762 p T^{6} - 638 p^{2} T^{7} + 186 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 3 T + 210 T^{2} + 331 T^{3} + 18870 T^{4} + 18021 T^{5} + 18870 p T^{6} + 331 p^{2} T^{7} + 210 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 23 T + 423 T^{2} + 5144 T^{3} + 54970 T^{4} + 448386 T^{5} + 54970 p T^{6} + 5144 p^{2} T^{7} + 423 p^{3} T^{8} + 23 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 15 T + 114 T^{2} + 769 T^{3} + 9652 T^{4} + 101401 T^{5} + 9652 p T^{6} + 769 p^{2} T^{7} + 114 p^{3} T^{8} + 15 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 18 T + 198 T^{2} - 842 T^{3} - 1242 T^{4} + 58612 T^{5} - 1242 p T^{6} - 842 p^{2} T^{7} + 198 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 12 T + 374 T^{2} + 3302 T^{3} + 54322 T^{4} + 348006 T^{5} + 54322 p T^{6} + 3302 p^{2} T^{7} + 374 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 43 T + 1030 T^{2} + 16873 T^{3} + 208660 T^{4} + 2006533 T^{5} + 208660 p T^{6} + 16873 p^{2} T^{7} + 1030 p^{3} T^{8} + 43 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 16 T + 459 T^{2} - 4992 T^{3} + 77114 T^{4} - 589280 T^{5} + 77114 p T^{6} - 4992 p^{2} T^{7} + 459 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 11 T + 239 T^{2} + 908 T^{3} + 14090 T^{4} - 13950 T^{5} + 14090 p T^{6} + 908 p^{2} T^{7} + 239 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 9 T + 292 T^{2} + 2533 T^{3} + 46058 T^{4} + 295137 T^{5} + 46058 p T^{6} + 2533 p^{2} T^{7} + 292 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 20 T + 513 T^{2} + 6480 T^{3} + 95726 T^{4} + 875640 T^{5} + 95726 p T^{6} + 6480 p^{2} T^{7} + 513 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
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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

−6.38695994345498654932656467692, −6.16662949394865413157622259756, −5.83931528728388982847793171298, −5.81601280222161358678295814315, −5.75804588814740925533631843653, −5.47699039765571605413702617627, −5.28131244456817007072907984505, −5.03397735198901837534351323794, −4.98106809944589618713419464844, −4.93771413603657145765520175811, −4.33697843072874445068502507500, −4.21137632431729740662498540110, −4.20356233661791631324312096502, −4.12498171055860450153421262354, −3.81419502257875743152068374570, −3.31068836199948105588713646681, −3.14694387835961369838843022621, −3.06080739337280028858120911177, −2.61043506732223336065097066882, −2.53778388717430804672205373606, −2.48305926292531706556335012734, −2.10394268663295716699422326766, −1.66865126958768523449486237516, −1.50458402205591738287703042178, −1.50377227904679219110838107341, 0, 0, 0, 0, 0, 1.50377227904679219110838107341, 1.50458402205591738287703042178, 1.66865126958768523449486237516, 2.10394268663295716699422326766, 2.48305926292531706556335012734, 2.53778388717430804672205373606, 2.61043506732223336065097066882, 3.06080739337280028858120911177, 3.14694387835961369838843022621, 3.31068836199948105588713646681, 3.81419502257875743152068374570, 4.12498171055860450153421262354, 4.20356233661791631324312096502, 4.21137632431729740662498540110, 4.33697843072874445068502507500, 4.93771413603657145765520175811, 4.98106809944589618713419464844, 5.03397735198901837534351323794, 5.28131244456817007072907984505, 5.47699039765571605413702617627, 5.75804588814740925533631843653, 5.81601280222161358678295814315, 5.83931528728388982847793171298, 6.16662949394865413157622259756, 6.38695994345498654932656467692

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