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Properties

Label 10-1449e5-1.1-c1e5-0-0
Degree $10$
Conductor $6.388\times 10^{15}$
Sign $1$
Analytic cond. $207361.$
Root an. cond. $3.40151$
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 + 3·4-s + 4·5-s + 5·7-s − 3·8-s − 8·10-s + 4·11-s − 6·13-s − 10·14-s − 16-s + 12·17-s + 6·19-s + 12·20-s − 8·22-s + 5·23-s + 5·25-s + 12·26-s + 15·28-s + 4·29-s + 30·31-s + 5·32-s − 24·34-s + 20·35-s + 4·37-s − 12·38-s − 12·40-s − 6·41-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 1.78·5-s + 1.88·7-s − 1.06·8-s − 2.52·10-s + 1.20·11-s − 1.66·13-s − 2.67·14-s − 1/4·16-s + 2.91·17-s + 1.37·19-s + 2.68·20-s − 1.70·22-s + 1.04·23-s + 25-s + 2.35·26-s + 2.83·28-s + 0.742·29-s + 5.38·31-s + 0.883·32-s − 4.11·34-s + 3.38·35-s + 0.657·37-s − 1.94·38-s − 1.89·40-s − 0.937·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 7^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 7^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{10} \cdot 7^{5} \cdot 23^{5}\)
Sign: $1$
Analytic conductor: \(207361.\)
Root analytic conductor: \(3.40151\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 3^{10} \cdot 7^{5} \cdot 23^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.831469306\)
\(L(\frac12)\) \(\approx\) \(7.831469306\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - T )^{5} \)
23$C_1$ \( ( 1 - T )^{5} \)
good2$C_2 \wr S_5$ \( 1 + p T + T^{2} - T^{3} + p T^{4} + 7 T^{5} + p^{2} T^{6} - p^{2} T^{7} + p^{3} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 - 4 T + 11 T^{2} - 26 T^{3} + 92 T^{4} - 228 T^{5} + 92 p T^{6} - 26 p^{2} T^{7} + 11 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 4 T + 27 T^{2} - 28 T^{3} + 126 T^{4} + 400 T^{5} + 126 p T^{6} - 28 p^{2} T^{7} + 27 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 6 T + 56 T^{2} + 266 T^{3} + 1351 T^{4} + 4944 T^{5} + 1351 p T^{6} + 266 p^{2} T^{7} + 56 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 12 T + 91 T^{2} - 430 T^{3} + 1692 T^{4} - 6148 T^{5} + 1692 p T^{6} - 430 p^{2} T^{7} + 91 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 6 T + 67 T^{2} - 360 T^{3} + 2334 T^{4} - 9220 T^{5} + 2334 p T^{6} - 360 p^{2} T^{7} + 67 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 4 T + 34 T^{2} - 214 T^{3} + 1217 T^{4} - 4232 T^{5} + 1217 p T^{6} - 214 p^{2} T^{7} + 34 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 30 T + 502 T^{2} - 5646 T^{3} + 46995 T^{4} - 297598 T^{5} + 46995 p T^{6} - 5646 p^{2} T^{7} + 502 p^{3} T^{8} - 30 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 4 T + 109 T^{2} - 216 T^{3} + 5126 T^{4} - 5064 T^{5} + 5126 p T^{6} - 216 p^{2} T^{7} + 109 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 6 T + 176 T^{2} + 838 T^{3} + 13295 T^{4} + 49000 T^{5} + 13295 p T^{6} + 838 p^{2} T^{7} + 176 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 12 T + 191 T^{2} + 1696 T^{3} + 16226 T^{4} + 101736 T^{5} + 16226 p T^{6} + 1696 p^{2} T^{7} + 191 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 10 T + 110 T^{2} - 38 T^{3} - 3609 T^{4} - 58894 T^{5} - 3609 p T^{6} - 38 p^{2} T^{7} + 110 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 16 T + 317 T^{2} + 3240 T^{3} + 35942 T^{4} + 254032 T^{5} + 35942 p T^{6} + 3240 p^{2} T^{7} + 317 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 22 T + 7 p T^{2} + 5194 T^{3} + 54600 T^{4} + 458288 T^{5} + 54600 p T^{6} + 5194 p^{2} T^{7} + 7 p^{4} T^{8} + 22 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 18 T + 339 T^{2} + 3954 T^{3} + 43744 T^{4} + 348488 T^{5} + 43744 p T^{6} + 3954 p^{2} T^{7} + 339 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 2 T + 35 T^{2} - 204 T^{3} + 6790 T^{4} + 16644 T^{5} + 6790 p T^{6} - 204 p^{2} T^{7} + 35 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 4 T + 254 T^{2} + 860 T^{3} + 30833 T^{4} + 86976 T^{5} + 30833 p T^{6} + 860 p^{2} T^{7} + 254 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 2 T + 168 T^{2} + 946 T^{3} + 18175 T^{4} + 89144 T^{5} + 18175 p T^{6} + 946 p^{2} T^{7} + 168 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 30 T + 703 T^{2} - 10676 T^{3} + 136374 T^{4} - 1310412 T^{5} + 136374 p T^{6} - 10676 p^{2} T^{7} + 703 p^{3} T^{8} - 30 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 8 T + 359 T^{2} + 2224 T^{3} + 55778 T^{4} + 264336 T^{5} + 55778 p T^{6} + 2224 p^{2} T^{7} + 359 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 20 T + 511 T^{2} - 6422 T^{3} + 92672 T^{4} - 821572 T^{5} + 92672 p T^{6} - 6422 p^{2} T^{7} + 511 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 12 T + 371 T^{2} + 3294 T^{3} + 61432 T^{4} + 417340 T^{5} + 61432 p T^{6} + 3294 p^{2} T^{7} + 371 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.97572633958835326963066484426, −5.53479416998887404198685294861, −5.28387919309721862778786081181, −5.23518501540830624627318337275, −4.95111425524653649791495248189, −4.78041150367619318643399276887, −4.62173243496883973223584413473, −4.56629306141409815455331006531, −4.51366669472755673491289283218, −4.49143852026301040558500622959, −3.54567545193528650777955646711, −3.43958663396641057493419974202, −3.43039159923026231303059761610, −3.11344738560897280253058616423, −3.02492429308283463717085006854, −2.71947553984394305517198844332, −2.57978377742495937801420634261, −2.12605179430903130044972774285, −1.94024711811846161341161892521, −1.92046405249632119270096136298, −1.39066671614041300995345901557, −1.36933971130617454660814374318, −1.06067370396726172783507812074, −1.01087951568853726922236975828, −0.48639957825520934789897714107, 0.48639957825520934789897714107, 1.01087951568853726922236975828, 1.06067370396726172783507812074, 1.36933971130617454660814374318, 1.39066671614041300995345901557, 1.92046405249632119270096136298, 1.94024711811846161341161892521, 2.12605179430903130044972774285, 2.57978377742495937801420634261, 2.71947553984394305517198844332, 3.02492429308283463717085006854, 3.11344738560897280253058616423, 3.43039159923026231303059761610, 3.43958663396641057493419974202, 3.54567545193528650777955646711, 4.49143852026301040558500622959, 4.51366669472755673491289283218, 4.56629306141409815455331006531, 4.62173243496883973223584413473, 4.78041150367619318643399276887, 4.95111425524653649791495248189, 5.23518501540830624627318337275, 5.28387919309721862778786081181, 5.53479416998887404198685294861, 5.97572633958835326963066484426

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