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Properties

Label 10-644e5-1.1-c1e5-0-0
Degree $10$
Conductor $1.108\times 10^{14}$
Sign $1$
Analytic cond. $3595.96$
Root an. cond. $2.26767$
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  + 3·3-s + 2·5-s − 5·7-s + 2·9-s + 2·11-s + 13·13-s + 6·15-s + 4·17-s + 12·19-s − 15·21-s − 5·23-s − 25-s − 27-s + 13·29-s − 3·31-s + 6·33-s − 10·35-s − 4·37-s + 39·39-s + 41-s − 8·43-s + 4·45-s + 5·47-s + 15·49-s + 12·51-s − 8·53-s + 4·55-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.894·5-s − 1.88·7-s + 2/3·9-s + 0.603·11-s + 3.60·13-s + 1.54·15-s + 0.970·17-s + 2.75·19-s − 3.27·21-s − 1.04·23-s − 1/5·25-s − 0.192·27-s + 2.41·29-s − 0.538·31-s + 1.04·33-s − 1.69·35-s − 0.657·37-s + 6.24·39-s + 0.156·41-s − 1.21·43-s + 0.596·45-s + 0.729·47-s + 15/7·49-s + 1.68·51-s − 1.09·53-s + 0.539·55-s + ⋯

Functional equation

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

Invariants

Degree: \(10\)
Conductor: \(2^{10} \cdot 7^{5} \cdot 23^{5}\)
Sign: $1$
Analytic conductor: \(3595.96\)
Root analytic conductor: \(2.26767\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{10} \cdot 7^{5} \cdot 23^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(9.798004877\)
\(L(\frac12)\) \(\approx\) \(9.798004877\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + T )^{5} \)
23$C_1$ \( ( 1 + T )^{5} \)
good3$C_2 \wr S_5$ \( 1 - p T + 7 T^{2} - 14 T^{3} + 34 T^{4} - 68 T^{5} + 34 p T^{6} - 14 p^{2} T^{7} + 7 p^{3} T^{8} - p^{5} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 - 2 T + p T^{2} - 6 T^{3} + 44 T^{4} - 24 p T^{5} + 44 p T^{6} - 6 p^{2} T^{7} + p^{4} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 2 T + 7 T^{2} - 20 T^{3} + 190 T^{4} - 316 T^{5} + 190 p T^{6} - 20 p^{2} T^{7} + 7 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - p T + 85 T^{2} - 356 T^{3} + 1110 T^{4} - 3438 T^{5} + 1110 p T^{6} - 356 p^{2} T^{7} + 85 p^{3} T^{8} - p^{5} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 4 T + 37 T^{2} - 202 T^{3} + 936 T^{4} - 4300 T^{5} + 936 p T^{6} - 202 p^{2} T^{7} + 37 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 12 T + 5 p T^{2} - 592 T^{3} + 3162 T^{4} - 14856 T^{5} + 3162 p T^{6} - 592 p^{2} T^{7} + 5 p^{4} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 13 T + 5 p T^{2} - 1204 T^{3} + 8338 T^{4} - 48762 T^{5} + 8338 p T^{6} - 1204 p^{2} T^{7} + 5 p^{4} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 3 T + 13 T^{2} - 198 T^{3} + 916 T^{4} + 2768 T^{5} + 916 p T^{6} - 198 p^{2} T^{7} + 13 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 4 T + 97 T^{2} + 80 T^{3} + 3682 T^{4} - 3880 T^{5} + 3682 p T^{6} + 80 p^{2} T^{7} + 97 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - T + 121 T^{2} - 68 T^{3} + 8206 T^{4} - 4246 T^{5} + 8206 p T^{6} - 68 p^{2} T^{7} + 121 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 8 T + 127 T^{2} + 352 T^{3} + 4338 T^{4} - 1296 T^{5} + 4338 p T^{6} + 352 p^{2} T^{7} + 127 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 5 T + 105 T^{2} - 298 T^{3} + 5648 T^{4} - 7120 T^{5} + 5648 p T^{6} - 298 p^{2} T^{7} + 105 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 8 T + 121 T^{2} + 752 T^{3} + 10234 T^{4} + 62992 T^{5} + 10234 p T^{6} + 752 p^{2} T^{7} + 121 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 12 T + 251 T^{2} - 2754 T^{3} + 27184 T^{4} - 241644 T^{5} + 27184 p T^{6} - 2754 p^{2} T^{7} + 251 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 20 T + 365 T^{2} - 4638 T^{3} + 47608 T^{4} - 416704 T^{5} + 47608 p T^{6} - 4638 p^{2} T^{7} + 365 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 12 T + 239 T^{2} + 1332 T^{3} + 17606 T^{4} + 61744 T^{5} + 17606 p T^{6} + 1332 p^{2} T^{7} + 239 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 9 T + 175 T^{2} - 332 T^{3} + 8806 T^{4} + 23754 T^{5} + 8806 p T^{6} - 332 p^{2} T^{7} + 175 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 9 T + 253 T^{2} + 2140 T^{3} + 32218 T^{4} + 217814 T^{5} + 32218 p T^{6} + 2140 p^{2} T^{7} + 253 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 8 T + 235 T^{2} + 620 T^{3} + 18278 T^{4} - 104 p T^{5} + 18278 p T^{6} + 620 p^{2} T^{7} + 235 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 28 T + 575 T^{2} + 8448 T^{3} + 102170 T^{4} + 1012488 T^{5} + 102170 p T^{6} + 8448 p^{2} T^{7} + 575 p^{3} T^{8} + 28 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 32 T + 593 T^{2} - 8038 T^{3} + 91908 T^{4} - 923620 T^{5} + 91908 p T^{6} - 8038 p^{2} T^{7} + 593 p^{3} T^{8} - 32 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 4 T + 289 T^{2} - 1438 T^{3} + 47792 T^{4} - 176000 T^{5} + 47792 p T^{6} - 1438 p^{2} T^{7} + 289 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
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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

−6.44392132234959473058426412852, −6.23627460997414417135387966788, −6.09288142992935337203432906932, −5.99291504974358417261356609866, −5.86592199269953802002697298053, −5.28430117269705697168936253973, −5.25793256535673690117212085756, −5.22969189092878489635355085946, −5.21424969637098265872416757539, −4.41891264819040350668245904011, −4.21252806272591130336908690675, −3.97100658097703822156604634605, −3.73422265729056473082784377557, −3.67152555126827602739945331809, −3.65349872272827095655780785260, −3.14433545721231089990199690398, −3.04340842656167423704613271672, −2.81190907382411647530473884729, −2.67281038292489146725014239221, −2.60411991707496156001830270983, −1.79632357507754833535704374709, −1.52056718611942454322189858960, −1.50492395757221088481247681802, −0.993744029390338225978274947159, −0.68321297020692240351439361056, 0.68321297020692240351439361056, 0.993744029390338225978274947159, 1.50492395757221088481247681802, 1.52056718611942454322189858960, 1.79632357507754833535704374709, 2.60411991707496156001830270983, 2.67281038292489146725014239221, 2.81190907382411647530473884729, 3.04340842656167423704613271672, 3.14433545721231089990199690398, 3.65349872272827095655780785260, 3.67152555126827602739945331809, 3.73422265729056473082784377557, 3.97100658097703822156604634605, 4.21252806272591130336908690675, 4.41891264819040350668245904011, 5.21424969637098265872416757539, 5.22969189092878489635355085946, 5.25793256535673690117212085756, 5.28430117269705697168936253973, 5.86592199269953802002697298053, 5.99291504974358417261356609866, 6.09288142992935337203432906932, 6.23627460997414417135387966788, 6.44392132234959473058426412852

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