Properties

Label 10-6027e5-1.1-c1e5-0-1
Degree $10$
Conductor $7.953\times 10^{18}$
Sign $-1$
Analytic cond. $2.58161\times 10^{8}$
Root an. cond. $6.93727$
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  + 3·2-s − 5·3-s + 3·4-s − 3·5-s − 15·6-s − 8-s + 15·9-s − 9·10-s + 4·11-s − 15·12-s − 5·13-s + 15·15-s − 6·16-s − 12·17-s + 45·18-s − 10·19-s − 9·20-s + 12·22-s + 9·23-s + 5·24-s − 10·25-s − 15·26-s − 35·27-s + 7·29-s + 45·30-s − 10·31-s − 6·32-s + ⋯
L(s)  = 1  + 2.12·2-s − 2.88·3-s + 3/2·4-s − 1.34·5-s − 6.12·6-s − 0.353·8-s + 5·9-s − 2.84·10-s + 1.20·11-s − 4.33·12-s − 1.38·13-s + 3.87·15-s − 3/2·16-s − 2.91·17-s + 10.6·18-s − 2.29·19-s − 2.01·20-s + 2.55·22-s + 1.87·23-s + 1.02·24-s − 2·25-s − 2.94·26-s − 6.73·27-s + 1.29·29-s + 8.21·30-s − 1.79·31-s − 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 7^{10} \cdot 41^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 7^{10} \cdot 41^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{5} \cdot 7^{10} \cdot 41^{5}\)
Sign: $-1$
Analytic conductor: \(2.58161\times 10^{8}\)
Root analytic conductor: \(6.93727\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 3^{5} \cdot 7^{10} \cdot 41^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad3$C_1$ \( ( 1 + T )^{5} \)
7 \( 1 \)
41$C_1$ \( ( 1 - T )^{5} \)
good2$C_2 \wr S_5$ \( 1 - 3 T + 3 p T^{2} - p^{3} T^{3} + 9 T^{4} - 9 T^{5} + 9 p T^{6} - p^{5} T^{7} + 3 p^{4} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + 3 T + 19 T^{2} + 42 T^{3} + 33 p T^{4} + 289 T^{5} + 33 p^{2} T^{6} + 42 p^{2} T^{7} + 19 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 4 T + 31 T^{2} - 80 T^{3} + 402 T^{4} - 904 T^{5} + 402 p T^{6} - 80 p^{2} T^{7} + 31 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 5 T + 55 T^{2} + 212 T^{3} + 1323 T^{4} + 3925 T^{5} + 1323 p T^{6} + 212 p^{2} T^{7} + 55 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 12 T + 133 T^{2} + 888 T^{3} + 5370 T^{4} + 23252 T^{5} + 5370 p T^{6} + 888 p^{2} T^{7} + 133 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 10 T + 101 T^{2} + 560 T^{3} + 3312 T^{4} + 13504 T^{5} + 3312 p T^{6} + 560 p^{2} T^{7} + 101 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 9 T + 91 T^{2} - 372 T^{3} + 2163 T^{4} - 6149 T^{5} + 2163 p T^{6} - 372 p^{2} T^{7} + 91 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 7 T + 81 T^{2} - 478 T^{3} + 3517 T^{4} - 599 p T^{5} + 3517 p T^{6} - 478 p^{2} T^{7} + 81 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 10 T + 161 T^{2} + 1192 T^{3} + 10136 T^{4} + 54728 T^{5} + 10136 p T^{6} + 1192 p^{2} T^{7} + 161 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 11 T + 187 T^{2} + 1302 T^{3} + 12769 T^{4} + 65209 T^{5} + 12769 p T^{6} + 1302 p^{2} T^{7} + 187 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 2 T + 127 T^{2} + 96 T^{3} + 8074 T^{4} + 2088 T^{5} + 8074 p T^{6} + 96 p^{2} T^{7} + 127 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 7 T + 187 T^{2} - 1104 T^{3} + 15309 T^{4} - 73191 T^{5} + 15309 p T^{6} - 1104 p^{2} T^{7} + 187 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 9 T + 243 T^{2} + 1688 T^{3} + 24799 T^{4} + 129409 T^{5} + 24799 p T^{6} + 1688 p^{2} T^{7} + 243 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 8 T + 275 T^{2} + 1636 T^{3} + 30950 T^{4} + 137708 T^{5} + 30950 p T^{6} + 1636 p^{2} T^{7} + 275 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 6 T + 7 T^{2} - 352 T^{3} + 4448 T^{4} - 23464 T^{5} + 4448 p T^{6} - 352 p^{2} T^{7} + 7 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 9 T + 279 T^{2} + 1844 T^{3} + 34149 T^{4} + 173765 T^{5} + 34149 p T^{6} + 1844 p^{2} T^{7} + 279 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 4 T + 241 T^{2} - 924 T^{3} + 28252 T^{4} - 95588 T^{5} + 28252 p T^{6} - 924 p^{2} T^{7} + 241 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 10 T + 363 T^{2} - 2820 T^{3} + 52916 T^{4} - 305136 T^{5} + 52916 p T^{6} - 2820 p^{2} T^{7} + 363 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 7 T + 275 T^{2} - 1000 T^{3} + 30985 T^{4} - 68483 T^{5} + 30985 p T^{6} - 1000 p^{2} T^{7} + 275 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 50 T + 1239 T^{2} + 20360 T^{3} + 252322 T^{4} + 2527052 T^{5} + 252322 p T^{6} + 20360 p^{2} T^{7} + 1239 p^{3} T^{8} + 50 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 8 T + 393 T^{2} + 2376 T^{3} + 64286 T^{4} + 295504 T^{5} + 64286 p T^{6} + 2376 p^{2} T^{7} + 393 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 11 T + 209 T^{2} - 1366 T^{3} + 30673 T^{4} - 226047 T^{5} + 30673 p T^{6} - 1366 p^{2} T^{7} + 209 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.97405780222789780379497890907, −4.97138593575141021698500028307, −4.75232904080527185088437523496, −4.66933086242759725527899102567, −4.66552361927819688642258323632, −4.29960162201455916281792297786, −4.20396291716412772201040066009, −4.13453840930588948154102285747, −4.11829112434434780096282437763, −4.11672226685035845174033175568, −3.62700659588397528587789714020, −3.57862782909946700058056215517, −3.48273486024420291885890007304, −3.23079740928152214343934846091, −3.04207655389958411689156324613, −2.65045990796400709823903374129, −2.54974235847051188771886601034, −2.33813419254822556875654056198, −2.22334214636461401085301451645, −1.84607349328444075552392676068, −1.79014450543595923552451435921, −1.72193057182262560725442338835, −1.17487031644348674910864798787, −0.988880262270492529501408885392, −0.943769022431447146803064647171, 0, 0, 0, 0, 0, 0.943769022431447146803064647171, 0.988880262270492529501408885392, 1.17487031644348674910864798787, 1.72193057182262560725442338835, 1.79014450543595923552451435921, 1.84607349328444075552392676068, 2.22334214636461401085301451645, 2.33813419254822556875654056198, 2.54974235847051188771886601034, 2.65045990796400709823903374129, 3.04207655389958411689156324613, 3.23079740928152214343934846091, 3.48273486024420291885890007304, 3.57862782909946700058056215517, 3.62700659588397528587789714020, 4.11672226685035845174033175568, 4.11829112434434780096282437763, 4.13453840930588948154102285747, 4.20396291716412772201040066009, 4.29960162201455916281792297786, 4.66552361927819688642258323632, 4.66933086242759725527899102567, 4.75232904080527185088437523496, 4.97138593575141021698500028307, 4.97405780222789780379497890907

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