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Properties

Label 10-4600e5-1.1-c1e5-0-4
Degree $10$
Conductor $2.060\times 10^{18}$
Sign $-1$
Analytic cond. $6.68612\times 10^{7}$
Root an. cond. $6.06062$
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  − 3·3-s + 7-s − 9-s − 4·11-s + 13-s − 5·17-s + 4·19-s − 3·21-s − 5·23-s + 13·27-s − 11·29-s + 4·31-s + 12·33-s − 6·37-s − 3·39-s − 8·41-s − 3·43-s − 2·47-s − 18·49-s + 15·51-s − 18·53-s − 12·57-s + 23·59-s − 26·61-s − 63-s − 3·67-s + 15·69-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.377·7-s − 1/3·9-s − 1.20·11-s + 0.277·13-s − 1.21·17-s + 0.917·19-s − 0.654·21-s − 1.04·23-s + 2.50·27-s − 2.04·29-s + 0.718·31-s + 2.08·33-s − 0.986·37-s − 0.480·39-s − 1.24·41-s − 0.457·43-s − 0.291·47-s − 2.57·49-s + 2.10·51-s − 2.47·53-s − 1.58·57-s + 2.99·59-s − 3.32·61-s − 0.125·63-s − 0.366·67-s + 1.80·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{15} \cdot 5^{10} \cdot 23^{5}\)
Sign: $-1$
Analytic conductor: \(6.68612\times 10^{7}\)
Root analytic conductor: \(6.06062\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{15} \cdot 5^{10} \cdot 23^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
5 \( 1 \)
23$C_1$ \( ( 1 + T )^{5} \)
good3$C_2 \wr S_5$ \( 1 + p T + 10 T^{2} + 20 T^{3} + 44 T^{4} + 76 T^{5} + 44 p T^{6} + 20 p^{2} T^{7} + 10 p^{3} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 - T + 19 T^{2} - 16 T^{3} + 30 p T^{4} - 158 T^{5} + 30 p^{2} T^{6} - 16 p^{2} T^{7} + 19 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 4 T + 38 T^{2} + 8 p T^{3} + 557 T^{4} + 960 T^{5} + 557 p T^{6} + 8 p^{3} T^{7} + 38 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - T + 28 T^{2} - 4 T^{3} + 568 T^{4} - 220 T^{5} + 568 p T^{6} - 4 p^{2} T^{7} + 28 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 5 T + 57 T^{2} + 224 T^{3} + 1390 T^{4} + 4758 T^{5} + 1390 p T^{6} + 224 p^{2} T^{7} + 57 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 4 T + 42 T^{2} - 40 T^{3} + 317 T^{4} + 1304 T^{5} + 317 p T^{6} - 40 p^{2} T^{7} + 42 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 11 T + 134 T^{2} + 970 T^{3} + 238 p T^{4} + 37502 T^{5} + 238 p^{2} T^{6} + 970 p^{2} T^{7} + 134 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 4 T + 80 T^{2} - 337 T^{3} + 3437 T^{4} - 12806 T^{5} + 3437 p T^{6} - 337 p^{2} T^{7} + 80 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 6 T + 45 T^{2} + 312 T^{3} + 2822 T^{4} + 12004 T^{5} + 2822 p T^{6} + 312 p^{2} T^{7} + 45 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 8 T + 153 T^{2} + 641 T^{3} + 8317 T^{4} + 23597 T^{5} + 8317 p T^{6} + 641 p^{2} T^{7} + 153 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 3 T + 71 T^{2} + 192 T^{3} + 4450 T^{4} + 13194 T^{5} + 4450 p T^{6} + 192 p^{2} T^{7} + 71 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 2 T + 68 T^{2} + 343 T^{3} + 3101 T^{4} + 29318 T^{5} + 3101 p T^{6} + 343 p^{2} T^{7} + 68 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 18 T + 213 T^{2} + 1184 T^{3} + 4462 T^{4} + 2236 T^{5} + 4462 p T^{6} + 1184 p^{2} T^{7} + 213 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 23 T + 348 T^{2} - 4341 T^{3} + 43633 T^{4} - 357240 T^{5} + 43633 p T^{6} - 4341 p^{2} T^{7} + 348 p^{3} T^{8} - 23 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 26 T + 541 T^{2} + 7192 T^{3} + 81278 T^{4} + 683676 T^{5} + 81278 p T^{6} + 7192 p^{2} T^{7} + 541 p^{3} T^{8} + 26 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 3 T + 275 T^{2} + 612 T^{3} + 33150 T^{4} + 55586 T^{5} + 33150 p T^{6} + 612 p^{2} T^{7} + 275 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 2 T + 288 T^{2} + 435 T^{3} + 36805 T^{4} + 41598 T^{5} + 36805 p T^{6} + 435 p^{2} T^{7} + 288 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 4 T - 3 T^{2} + 167 T^{3} + 6489 T^{4} + 34251 T^{5} + 6489 p T^{6} + 167 p^{2} T^{7} - 3 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 43 T + 1029 T^{2} - 16880 T^{3} + 210588 T^{4} - 2084442 T^{5} + 210588 p T^{6} - 16880 p^{2} T^{7} + 1029 p^{3} T^{8} - 43 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 30 T + 584 T^{2} + 8400 T^{3} + 102155 T^{4} + 1014020 T^{5} + 102155 p T^{6} + 8400 p^{2} T^{7} + 584 p^{3} T^{8} + 30 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 15 T + 291 T^{2} - 2152 T^{3} + 25484 T^{4} - 1554 p T^{5} + 25484 p T^{6} - 2152 p^{2} T^{7} + 291 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 8 T + 257 T^{2} + 1592 T^{3} + 32590 T^{4} + 159328 T^{5} + 32590 p T^{6} + 1592 p^{2} T^{7} + 257 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.28917568834485948457583763101, −5.15940940147053003680079585787, −5.06465895440179082554051490118, −4.87849080533823714397843837810, −4.84498876579350754775107016715, −4.59846637137409192373723603804, −4.58350036803510447691693075582, −4.17568980098997419503665195285, −4.13363759701252216263286449543, −3.83263244786778825739887429230, −3.61473123589324220854371874623, −3.39325117997997099877192420350, −3.32105983179792084302241985968, −3.29721860954859952353272385735, −3.24235432881110504975201712335, −2.60503488645922609184985183492, −2.50670749626134397358955452780, −2.39409358524364160843291578051, −2.38079266688369237790825097893, −2.13185409700248642782152472093, −1.69926867271656180868227390547, −1.58199539642639426633881673353, −1.25684170420085087120493260866, −1.13492912378552764650173446555, −1.10026395080106379606704945980, 0, 0, 0, 0, 0, 1.10026395080106379606704945980, 1.13492912378552764650173446555, 1.25684170420085087120493260866, 1.58199539642639426633881673353, 1.69926867271656180868227390547, 2.13185409700248642782152472093, 2.38079266688369237790825097893, 2.39409358524364160843291578051, 2.50670749626134397358955452780, 2.60503488645922609184985183492, 3.24235432881110504975201712335, 3.29721860954859952353272385735, 3.32105983179792084302241985968, 3.39325117997997099877192420350, 3.61473123589324220854371874623, 3.83263244786778825739887429230, 4.13363759701252216263286449543, 4.17568980098997419503665195285, 4.58350036803510447691693075582, 4.59846637137409192373723603804, 4.84498876579350754775107016715, 4.87849080533823714397843837810, 5.06465895440179082554051490118, 5.15940940147053003680079585787, 5.28917568834485948457583763101

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