Properties

Label 10-2960e5-1.1-c1e5-0-1
Degree $10$
Conductor $2.272\times 10^{17}$
Sign $1$
Analytic cond. $7.37639\times 10^{6}$
Root an. cond. $4.86165$
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  + 3-s − 5·5-s + 7-s − 6·9-s + 7·11-s + 4·13-s − 5·15-s + 10·19-s + 21-s + 8·23-s + 15·25-s − 5·27-s − 8·29-s + 2·31-s + 7·33-s − 5·35-s + 5·37-s + 4·39-s − 13·41-s + 18·43-s + 30·45-s + 9·47-s − 20·49-s − 13·53-s − 35·55-s + 10·57-s + 24·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 2.23·5-s + 0.377·7-s − 2·9-s + 2.11·11-s + 1.10·13-s − 1.29·15-s + 2.29·19-s + 0.218·21-s + 1.66·23-s + 3·25-s − 0.962·27-s − 1.48·29-s + 0.359·31-s + 1.21·33-s − 0.845·35-s + 0.821·37-s + 0.640·39-s − 2.03·41-s + 2.74·43-s + 4.47·45-s + 1.31·47-s − 2.85·49-s − 1.78·53-s − 4.71·55-s + 1.32·57-s + 3.12·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{5} \cdot 37^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{5} \cdot 37^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{20} \cdot 5^{5} \cdot 37^{5}\)
Sign: $1$
Analytic conductor: \(7.37639\times 10^{6}\)
Root analytic conductor: \(4.86165\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{20} \cdot 5^{5} \cdot 37^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.696381483\)
\(L(\frac12)\) \(\approx\) \(7.696381483\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{5} \)
37$C_1$ \( ( 1 - T )^{5} \)
good3$C_2 \wr S_5$ \( 1 - T + 7 T^{2} - 8 T^{3} + 26 T^{4} - 28 T^{5} + 26 p T^{6} - 8 p^{2} T^{7} + 7 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 - T + 3 p T^{2} - 20 T^{3} + 32 p T^{4} - 176 T^{5} + 32 p^{2} T^{6} - 20 p^{2} T^{7} + 3 p^{4} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 7 T + 35 T^{2} - 116 T^{3} + 34 p T^{4} - 1146 T^{5} + 34 p^{2} T^{6} - 116 p^{2} T^{7} + 35 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 4 T + 37 T^{2} - 36 T^{3} + 314 T^{4} + 560 T^{5} + 314 p T^{6} - 36 p^{2} T^{7} + 37 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 57 T^{2} - 20 T^{3} + 1578 T^{4} - 568 T^{5} + 1578 p T^{6} - 20 p^{2} T^{7} + 57 p^{3} T^{8} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 10 T + 69 T^{2} - 10 p T^{3} + 246 T^{4} + 1544 T^{5} + 246 p T^{6} - 10 p^{3} T^{7} + 69 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 8 T + 107 T^{2} - 608 T^{3} + 4626 T^{4} - 19696 T^{5} + 4626 p T^{6} - 608 p^{2} T^{7} + 107 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 8 T + 113 T^{2} + 688 T^{3} + 5674 T^{4} + 27344 T^{5} + 5674 p T^{6} + 688 p^{2} T^{7} + 113 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 2 T + 25 T^{2} - 234 T^{3} + 1490 T^{4} - 3108 T^{5} + 1490 p T^{6} - 234 p^{2} T^{7} + 25 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 13 T + 165 T^{2} + 36 p T^{3} + 12354 T^{4} + 77550 T^{5} + 12354 p T^{6} + 36 p^{3} T^{7} + 165 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 18 T + 255 T^{2} - 2232 T^{3} + 18130 T^{4} - 116332 T^{5} + 18130 p T^{6} - 2232 p^{2} T^{7} + 255 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 9 T + 181 T^{2} - 1020 T^{3} + 12964 T^{4} - 55232 T^{5} + 12964 p T^{6} - 1020 p^{2} T^{7} + 181 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 13 T + 249 T^{2} + 1948 T^{3} + 21978 T^{4} + 128798 T^{5} + 21978 p T^{6} + 1948 p^{2} T^{7} + 249 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 24 T + 333 T^{2} - 3754 T^{3} + 37854 T^{4} - 319940 T^{5} + 37854 p T^{6} - 3754 p^{2} T^{7} + 333 p^{3} T^{8} - 24 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 2 T + 89 T^{2} - 456 T^{3} + 6338 T^{4} - 27092 T^{5} + 6338 p T^{6} - 456 p^{2} T^{7} + 89 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 22 T + 425 T^{2} - 5058 T^{3} + 56750 T^{4} - 472312 T^{5} + 56750 p T^{6} - 5058 p^{2} T^{7} + 425 p^{3} T^{8} - 22 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 21 T + 455 T^{2} - 5436 T^{3} + 66334 T^{4} - 549118 T^{5} + 66334 p T^{6} - 5436 p^{2} T^{7} + 455 p^{3} T^{8} - 21 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 29 T + 629 T^{2} - 8988 T^{3} + 108194 T^{4} - 996078 T^{5} + 108194 p T^{6} - 8988 p^{2} T^{7} + 629 p^{3} T^{8} - 29 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 6 T + 317 T^{2} - 1878 T^{3} + 44638 T^{4} - 221140 T^{5} + 44638 p T^{6} - 1878 p^{2} T^{7} + 317 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 33 T + 783 T^{2} - 12460 T^{3} + 161058 T^{4} - 1607612 T^{5} + 161058 p T^{6} - 12460 p^{2} T^{7} + 783 p^{3} T^{8} - 33 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 26 T + 485 T^{2} + 6840 T^{3} + 84402 T^{4} + 849564 T^{5} + 84402 p T^{6} + 6840 p^{2} T^{7} + 485 p^{3} T^{8} + 26 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 26 T + 605 T^{2} - 9176 T^{3} + 122610 T^{4} - 1281852 T^{5} + 122610 p T^{6} - 9176 p^{2} T^{7} + 605 p^{3} T^{8} - 26 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.14531779902586907859219822815, −5.11110270182981155801238143565, −4.89916333012288435473050949059, −4.72155712373662968952110595066, −4.45647386045317901585884069422, −4.27113954283555978883193874720, −4.10831661945571945661652163096, −3.85074813720240732458253752171, −3.75135088626877643288320754527, −3.70221094113659382478610702057, −3.34558979732480648330080834084, −3.34228027237675871998777035698, −3.11779423735335569896339903804, −3.09727284414516070301063390190, −3.09196447056445078476904655830, −2.32051158214688353618926143789, −2.28741462417006036445843477260, −2.14855211245683531486224272185, −2.14552724634703543335637460282, −1.50302268950116795340134102381, −1.20427433760277033451521643482, −0.976268763879248203904474414928, −0.916375578361149214345524148066, −0.60531509951207673363105434384, −0.42799123564463035723194649151, 0.42799123564463035723194649151, 0.60531509951207673363105434384, 0.916375578361149214345524148066, 0.976268763879248203904474414928, 1.20427433760277033451521643482, 1.50302268950116795340134102381, 2.14552724634703543335637460282, 2.14855211245683531486224272185, 2.28741462417006036445843477260, 2.32051158214688353618926143789, 3.09196447056445078476904655830, 3.09727284414516070301063390190, 3.11779423735335569896339903804, 3.34228027237675871998777035698, 3.34558979732480648330080834084, 3.70221094113659382478610702057, 3.75135088626877643288320754527, 3.85074813720240732458253752171, 4.10831661945571945661652163096, 4.27113954283555978883193874720, 4.45647386045317901585884069422, 4.72155712373662968952110595066, 4.89916333012288435473050949059, 5.11110270182981155801238143565, 5.14531779902586907859219822815

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