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Properties

Label 10-6864e5-1.1-c1e5-0-0
Degree $10$
Conductor $1.524\times 10^{19}$
Sign $1$
Analytic cond. $4.94620\times 10^{8}$
Root an. cond. $7.40333$
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  − 5·3-s + 5-s + 3·7-s + 15·9-s − 5·11-s + 5·13-s − 5·15-s + 8·17-s − 10·19-s − 15·21-s + 7·23-s − 4·25-s − 35·27-s − 3·29-s + 4·31-s + 25·33-s + 3·35-s + 6·37-s − 25·39-s + 13·41-s − 3·43-s + 15·45-s − 2·47-s − 8·49-s − 40·51-s + 4·53-s − 5·55-s + ⋯
L(s)  = 1  − 2.88·3-s + 0.447·5-s + 1.13·7-s + 5·9-s − 1.50·11-s + 1.38·13-s − 1.29·15-s + 1.94·17-s − 2.29·19-s − 3.27·21-s + 1.45·23-s − 4/5·25-s − 6.73·27-s − 0.557·29-s + 0.718·31-s + 4.35·33-s + 0.507·35-s + 0.986·37-s − 4.00·39-s + 2.03·41-s − 0.457·43-s + 2.23·45-s − 0.291·47-s − 8/7·49-s − 5.60·51-s + 0.549·53-s − 0.674·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{5} \cdot 11^{5} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{5} \cdot 11^{5} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{20} \cdot 3^{5} \cdot 11^{5} \cdot 13^{5}\)
Sign: $1$
Analytic conductor: \(4.94620\times 10^{8}\)
Root analytic conductor: \(7.40333\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{20} \cdot 3^{5} \cdot 11^{5} \cdot 13^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.192760817\)
\(L(\frac12)\) \(\approx\) \(4.192760817\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 + T )^{5} \)
11$C_1$ \( ( 1 + T )^{5} \)
13$C_1$ \( ( 1 - T )^{5} \)
good5$C_2 \wr S_5$ \( 1 - T + p T^{2} - 12 T^{3} + 4 p T^{4} - 42 T^{5} + 4 p^{2} T^{6} - 12 p^{2} T^{7} + p^{4} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 - 3 T + 17 T^{2} - 60 T^{3} + 164 T^{4} - 530 T^{5} + 164 p T^{6} - 60 p^{2} T^{7} + 17 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 8 T + 65 T^{2} - 342 T^{3} + 1814 T^{4} - 7452 T^{5} + 1814 p T^{6} - 342 p^{2} T^{7} + 65 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 10 T + 79 T^{2} + 340 T^{3} + 1466 T^{4} + 4916 T^{5} + 1466 p T^{6} + 340 p^{2} T^{7} + 79 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 7 T + 113 T^{2} - 568 T^{3} + 5068 T^{4} - 18778 T^{5} + 5068 p T^{6} - 568 p^{2} T^{7} + 113 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 3 T - 35 T^{2} - 282 T^{3} + 644 T^{4} + 10066 T^{5} + 644 p T^{6} - 282 p^{2} T^{7} - 35 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 4 T + 85 T^{2} - 410 T^{3} + 4528 T^{4} - 15716 T^{5} + 4528 p T^{6} - 410 p^{2} T^{7} + 85 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 6 T + 113 T^{2} - 696 T^{3} + 6530 T^{4} - 34564 T^{5} + 6530 p T^{6} - 696 p^{2} T^{7} + 113 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 13 T + 251 T^{2} - 2152 T^{3} + 22384 T^{4} - 132686 T^{5} + 22384 p T^{6} - 2152 p^{2} T^{7} + 251 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 3 T + 97 T^{2} + 78 T^{3} + 5942 T^{4} + 4798 T^{5} + 5942 p T^{6} + 78 p^{2} T^{7} + 97 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 2 T + 79 T^{2} - 16 T^{3} + 4446 T^{4} + 3996 T^{5} + 4446 p T^{6} - 16 p^{2} T^{7} + 79 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 4 T + 153 T^{2} - 240 T^{3} + 10106 T^{4} - 3096 T^{5} + 10106 p T^{6} - 240 p^{2} T^{7} + 153 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 21 T + 327 T^{2} + 3588 T^{3} + 37418 T^{4} + 305278 T^{5} + 37418 p T^{6} + 3588 p^{2} T^{7} + 327 p^{3} T^{8} + 21 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 15 T + 263 T^{2} + 2924 T^{3} + 29220 T^{4} + 245322 T^{5} + 29220 p T^{6} + 2924 p^{2} T^{7} + 263 p^{3} T^{8} + 15 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 9 T + 347 T^{2} + 2356 T^{3} + 47212 T^{4} + 234958 T^{5} + 47212 p T^{6} + 2356 p^{2} T^{7} + 347 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 4 T + 59 T^{2} + 528 T^{3} + 2162 T^{4} + 24360 T^{5} + 2162 p T^{6} + 528 p^{2} T^{7} + 59 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 19 T + 393 T^{2} - 4612 T^{3} + 56962 T^{4} - 478562 T^{5} + 56962 p T^{6} - 4612 p^{2} T^{7} + 393 p^{3} T^{8} - 19 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 8 T + 83 T^{2} + 534 T^{3} + 3678 T^{4} - 36540 T^{5} + 3678 p T^{6} + 534 p^{2} T^{7} + 83 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 2 T + 335 T^{2} - 600 T^{3} + 50090 T^{4} - 71148 T^{5} + 50090 p T^{6} - 600 p^{2} T^{7} + 335 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 12 T + 155 T^{2} - 1174 T^{3} + 21464 T^{4} - 183572 T^{5} + 21464 p T^{6} - 1174 p^{2} T^{7} + 155 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 32 T + 745 T^{2} - 12400 T^{3} + 164374 T^{4} - 1795360 T^{5} + 164374 p T^{6} - 12400 p^{2} T^{7} + 745 p^{3} T^{8} - 32 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.80876197838574037183219770964, −4.61787749975625215054643421606, −4.40830513081362643760149604752, −4.31475126239089763253263674693, −4.29889896091032569546861571460, −4.07645407822495013828101879461, −3.71710814530913938887336401668, −3.54805283429145511960274126488, −3.49252759749026636099666544461, −3.44899099254231110889819071055, −2.98691586252601523140170014228, −2.96376126291767695773861200308, −2.61382944890224788650920254675, −2.56636011080361752774302891659, −2.48190787809791536220189972926, −1.80431306609373019344988553028, −1.79041772105885891103790974200, −1.68585932633609670389577207378, −1.64614256682895311847104715469, −1.64253520715635596788854387636, −0.943365270347900650539184989146, −0.860172775804968814677130147434, −0.65290911447088810019641772735, −0.49991290604070523834266395960, −0.35521481485018965449200325244, 0.35521481485018965449200325244, 0.49991290604070523834266395960, 0.65290911447088810019641772735, 0.860172775804968814677130147434, 0.943365270347900650539184989146, 1.64253520715635596788854387636, 1.64614256682895311847104715469, 1.68585932633609670389577207378, 1.79041772105885891103790974200, 1.80431306609373019344988553028, 2.48190787809791536220189972926, 2.56636011080361752774302891659, 2.61382944890224788650920254675, 2.96376126291767695773861200308, 2.98691586252601523140170014228, 3.44899099254231110889819071055, 3.49252759749026636099666544461, 3.54805283429145511960274126488, 3.71710814530913938887336401668, 4.07645407822495013828101879461, 4.29889896091032569546861571460, 4.31475126239089763253263674693, 4.40830513081362643760149604752, 4.61787749975625215054643421606, 4.80876197838574037183219770964

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