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Properties

Label 10-1148e5-1.1-c1e5-0-2
Degree $10$
Conductor $1.994\times 10^{15}$
Sign $-1$
Analytic cond. $64728.4$
Root an. cond. $3.02767$
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  − 2·3-s − 5-s − 5·7-s − 5·9-s − 2·11-s − 13-s + 2·15-s − 3·17-s − 4·19-s + 10·21-s − 8·23-s − 15·25-s + 12·27-s − 9·29-s − 11·31-s + 4·33-s + 5·35-s − 11·37-s + 2·39-s + 5·41-s − 27·43-s + 5·45-s − 3·47-s + 15·49-s + 6·51-s − 19·53-s + 2·55-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s − 1.88·7-s − 5/3·9-s − 0.603·11-s − 0.277·13-s + 0.516·15-s − 0.727·17-s − 0.917·19-s + 2.18·21-s − 1.66·23-s − 3·25-s + 2.30·27-s − 1.67·29-s − 1.97·31-s + 0.696·33-s + 0.845·35-s − 1.80·37-s + 0.320·39-s + 0.780·41-s − 4.11·43-s + 0.745·45-s − 0.437·47-s + 15/7·49-s + 0.840·51-s − 2.60·53-s + 0.269·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 7^{5} \cdot 41^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 7^{5} \cdot 41^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{10} \cdot 7^{5} \cdot 41^{5}\)
Sign: $-1$
Analytic conductor: \(64728.4\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{10} \cdot 7^{5} \cdot 41^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
7$C_1$ \( ( 1 + T )^{5} \)
41$C_1$ \( ( 1 - T )^{5} \)
good3$C_2 \wr S_5$ \( 1 + 2 T + p^{2} T^{2} + 16 T^{3} + 43 T^{4} + 59 T^{5} + 43 p T^{6} + 16 p^{2} T^{7} + p^{5} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + T + 16 T^{2} + 12 T^{3} + 127 T^{4} + 78 T^{5} + 127 p T^{6} + 12 p^{2} T^{7} + 16 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 2 T + 26 T^{2} - 10 T^{3} + 15 p T^{4} - 728 T^{5} + 15 p^{2} T^{6} - 10 p^{2} T^{7} + 26 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + T + 42 T^{2} + 4 T^{3} + 835 T^{4} - 131 T^{5} + 835 p T^{6} + 4 p^{2} T^{7} + 42 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 3 T + 56 T^{2} + 126 T^{3} + 1591 T^{4} + 3013 T^{5} + 1591 p T^{6} + 126 p^{2} T^{7} + 56 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 4 T + 39 T^{2} + 48 T^{3} + 441 T^{4} - 365 T^{5} + 441 p T^{6} + 48 p^{2} T^{7} + 39 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 8 T + 83 T^{2} + 574 T^{3} + 3355 T^{4} + 18011 T^{5} + 3355 p T^{6} + 574 p^{2} T^{7} + 83 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 9 T + 68 T^{2} + 24 T^{3} - 1301 T^{4} - 16106 T^{5} - 1301 p T^{6} + 24 p^{2} T^{7} + 68 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 11 T + 112 T^{2} + 656 T^{3} + 5003 T^{4} + 25610 T^{5} + 5003 p T^{6} + 656 p^{2} T^{7} + 112 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 11 T + 145 T^{2} + 663 T^{3} + 4956 T^{4} + 12852 T^{5} + 4956 p T^{6} + 663 p^{2} T^{7} + 145 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 27 T + 382 T^{2} + 3986 T^{3} + 34437 T^{4} + 248301 T^{5} + 34437 p T^{6} + 3986 p^{2} T^{7} + 382 p^{3} T^{8} + 27 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 3 T + 167 T^{2} + 461 T^{3} + 13280 T^{4} + 29348 T^{5} + 13280 p T^{6} + 461 p^{2} T^{7} + 167 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 19 T + 298 T^{2} + 2774 T^{3} + 25329 T^{4} + 175326 T^{5} + 25329 p T^{6} + 2774 p^{2} T^{7} + 298 p^{3} T^{8} + 19 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 15 T + 316 T^{2} + 3050 T^{3} + 36379 T^{4} + 253030 T^{5} + 36379 p T^{6} + 3050 p^{2} T^{7} + 316 p^{3} T^{8} + 15 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 190 T^{2} - 30 T^{3} + 18957 T^{4} - 5364 T^{5} + 18957 p T^{6} - 30 p^{2} T^{7} + 190 p^{3} T^{8} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 21 T + 310 T^{2} + 3174 T^{3} + 31197 T^{4} + 258962 T^{5} + 31197 p T^{6} + 3174 p^{2} T^{7} + 310 p^{3} T^{8} + 21 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 16 T + 338 T^{2} + 4312 T^{3} + 46713 T^{4} + 451448 T^{5} + 46713 p T^{6} + 4312 p^{2} T^{7} + 338 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 10 T + 222 T^{2} + 2492 T^{3} + 25941 T^{4} + 257572 T^{5} + 25941 p T^{6} + 2492 p^{2} T^{7} + 222 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 14 T + 315 T^{2} + 3176 T^{3} + 45082 T^{4} + 346676 T^{5} + 45082 p T^{6} + 3176 p^{2} T^{7} + 315 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 2 T + 114 T^{2} + 338 T^{3} + 12089 T^{4} + 78912 T^{5} + 12089 p T^{6} + 338 p^{2} T^{7} + 114 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 6 T + 305 T^{2} - 1284 T^{3} + 45117 T^{4} - 153431 T^{5} + 45117 p T^{6} - 1284 p^{2} T^{7} + 305 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 20 T + 207 T^{2} - 2236 T^{3} + 33747 T^{4} - 423407 T^{5} + 33747 p T^{6} - 2236 p^{2} T^{7} + 207 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} \)
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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

−6.13683345106680733233481946381, −6.08292868938176301936981923247, −6.05208525202265636199626020314, −5.95967165021279489441086033902, −5.90724539023125577759734534227, −5.32067625999185495944569383974, −5.29698381822689661785406536059, −5.28817945300475455619323261394, −5.11610541204964368398134841012, −4.80117664371586015201548189838, −4.42444775792775362180891023346, −4.29043642749220374124490282993, −4.04774040221638631612231651313, −3.86908673936241250960549562459, −3.84440341959385043855013719783, −3.33246321303691667562510647431, −3.24518233323989573112435292222, −3.16302342806536617160179601269, −2.88719552588742212033633387253, −2.85705710765587692759979683619, −2.12971895010954271476658936940, −2.10636951313438139709708594409, −1.81257691335166189049675046466, −1.73648919498288764681241678112, −1.39605086295081559782390318399, 0, 0, 0, 0, 0, 1.39605086295081559782390318399, 1.73648919498288764681241678112, 1.81257691335166189049675046466, 2.10636951313438139709708594409, 2.12971895010954271476658936940, 2.85705710765587692759979683619, 2.88719552588742212033633387253, 3.16302342806536617160179601269, 3.24518233323989573112435292222, 3.33246321303691667562510647431, 3.84440341959385043855013719783, 3.86908673936241250960549562459, 4.04774040221638631612231651313, 4.29043642749220374124490282993, 4.42444775792775362180891023346, 4.80117664371586015201548189838, 5.11610541204964368398134841012, 5.28817945300475455619323261394, 5.29698381822689661785406536059, 5.32067625999185495944569383974, 5.90724539023125577759734534227, 5.95967165021279489441086033902, 6.05208525202265636199626020314, 6.08292868938176301936981923247, 6.13683345106680733233481946381

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