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Properties

Label 10-4026e5-1.1-c1e5-0-1
Degree $10$
Conductor $1.058\times 10^{18}$
Sign $1$
Analytic cond. $3.43363\times 10^{7}$
Root an. cond. $5.66990$
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  + 5·2-s − 5·3-s + 15·4-s + 7·5-s − 25·6-s + 35·8-s + 15·9-s + 35·10-s + 5·11-s − 75·12-s − 2·13-s − 35·15-s + 70·16-s + 2·17-s + 75·18-s + 18·19-s + 105·20-s + 25·22-s − 23-s − 175·24-s + 15·25-s − 10·26-s − 35·27-s + 7·29-s − 175·30-s + 126·32-s − 25·33-s + ⋯
L(s)  = 1  + 3.53·2-s − 2.88·3-s + 15/2·4-s + 3.13·5-s − 10.2·6-s + 12.3·8-s + 5·9-s + 11.0·10-s + 1.50·11-s − 21.6·12-s − 0.554·13-s − 9.03·15-s + 35/2·16-s + 0.485·17-s + 17.6·18-s + 4.12·19-s + 23.4·20-s + 5.33·22-s − 0.208·23-s − 35.7·24-s + 3·25-s − 1.96·26-s − 6.73·27-s + 1.29·29-s − 31.9·30-s + 22.2·32-s − 4.35·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{5} \cdot 11^{5} \cdot 61^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{5} \cdot 11^{5} \cdot 61^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{5} \cdot 3^{5} \cdot 11^{5} \cdot 61^{5}\)
Sign: $1$
Analytic conductor: \(3.43363\times 10^{7}\)
Root analytic conductor: \(5.66990\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{5} \cdot 3^{5} \cdot 11^{5} \cdot 61^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(178.7017536\)
\(L(\frac12)\) \(\approx\) \(178.7017536\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{5} \)
3$C_1$ \( ( 1 + T )^{5} \)
11$C_1$ \( ( 1 - T )^{5} \)
61$C_1$ \( ( 1 - T )^{5} \)
good5$C_2 \wr S_5$ \( 1 - 7 T + 34 T^{2} - 122 T^{3} + 363 T^{4} - 877 T^{5} + 363 p T^{6} - 122 p^{2} T^{7} + 34 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 2 p T^{2} + 3 T^{3} + 99 T^{4} + 10 T^{5} + 99 p T^{6} + 3 p^{2} T^{7} + 2 p^{4} T^{8} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 2 T + 34 T^{2} + 81 T^{3} + 724 T^{4} + 1272 T^{5} + 724 p T^{6} + 81 p^{2} T^{7} + 34 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 2 T + 53 T^{2} - 110 T^{3} + 1457 T^{4} - 2648 T^{5} + 1457 p T^{6} - 110 p^{2} T^{7} + 53 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 18 T + 200 T^{2} - 1571 T^{3} + 9599 T^{4} - 46478 T^{5} + 9599 p T^{6} - 1571 p^{2} T^{7} + 200 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + T - 4 T^{2} + 200 T^{3} + 740 T^{4} - 1946 T^{5} + 740 p T^{6} + 200 p^{2} T^{7} - 4 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 7 T + 130 T^{2} - 728 T^{3} + 7066 T^{4} - 30476 T^{5} + 7066 p T^{6} - 728 p^{2} T^{7} + 130 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 100 T^{2} - 67 T^{3} + 5228 T^{4} - 2620 T^{5} + 5228 p T^{6} - 67 p^{2} T^{7} + 100 p^{3} T^{8} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 11 T + 124 T^{2} - 916 T^{3} + 7078 T^{4} - 39344 T^{5} + 7078 p T^{6} - 916 p^{2} T^{7} + 124 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 8 T + 97 T^{2} - 558 T^{3} + 6329 T^{4} - 36766 T^{5} + 6329 p T^{6} - 558 p^{2} T^{7} + 97 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 7 T + 180 T^{2} + 1100 T^{3} + 14130 T^{4} + 69078 T^{5} + 14130 p T^{6} + 1100 p^{2} T^{7} + 180 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - T + 73 T^{2} + 3 p T^{3} + 1206 T^{4} + 19128 T^{5} + 1206 p T^{6} + 3 p^{3} T^{7} + 73 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 10 T + 144 T^{2} - 1327 T^{3} + 13765 T^{4} - 89890 T^{5} + 13765 p T^{6} - 1327 p^{2} T^{7} + 144 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 8 T + 228 T^{2} - 1307 T^{3} + 23296 T^{4} - 104348 T^{5} + 23296 p T^{6} - 1307 p^{2} T^{7} + 228 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 9 T + 331 T^{2} + 2311 T^{3} + 658 p T^{4} + 229080 T^{5} + 658 p^{2} T^{6} + 2311 p^{2} T^{7} + 331 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 34 T + 622 T^{2} - 8033 T^{3} + 82242 T^{4} - 728020 T^{5} + 82242 p T^{6} - 8033 p^{2} T^{7} + 622 p^{3} T^{8} - 34 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 13 T + 170 T^{2} - 2170 T^{3} + 22014 T^{4} - 162898 T^{5} + 22014 p T^{6} - 2170 p^{2} T^{7} + 170 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 15 T + 349 T^{2} - 3743 T^{3} + 51768 T^{4} - 414980 T^{5} + 51768 p T^{6} - 3743 p^{2} T^{7} + 349 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 27 T + 376 T^{2} + 3000 T^{3} + 16368 T^{4} + 90878 T^{5} + 16368 p T^{6} + 3000 p^{2} T^{7} + 376 p^{3} T^{8} + 27 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 11 T + 286 T^{2} - 2392 T^{3} + 41643 T^{4} - 296243 T^{5} + 41643 p T^{6} - 2392 p^{2} T^{7} + 286 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 37 T + 838 T^{2} - 13892 T^{3} + 182723 T^{4} - 1965609 T^{5} + 182723 p T^{6} - 13892 p^{2} T^{7} + 838 p^{3} T^{8} - 37 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.91977418794607829188825584376, −4.89820870246500834362846844918, −4.86805007066755470122665107195, −4.81136077351872981491203586629, −4.60241423104244304136854819574, −4.22467780332994002063946383692, −3.94282062098976836028335362510, −3.92480523196346363651629332528, −3.84128805810088760924826044197, −3.79002237117560914939173803414, −3.28999834391476005025175943919, −3.21187924115478014913971282982, −2.99485979106395518628171341060, −2.85255054181356782113057947793, −2.83411440003332910370433401340, −2.20223192475796944584197385901, −2.10955631557434139874069906457, −2.02077643168931595756146249273, −1.89759716170739343492373464596, −1.87530449815245068111317318944, −1.19696192855130500725328902994, −1.07137586310736095381193208687, −1.04001117995748847437104918688, −0.975816149364099101108131808012, −0.55802501252205784211459793584, 0.55802501252205784211459793584, 0.975816149364099101108131808012, 1.04001117995748847437104918688, 1.07137586310736095381193208687, 1.19696192855130500725328902994, 1.87530449815245068111317318944, 1.89759716170739343492373464596, 2.02077643168931595756146249273, 2.10955631557434139874069906457, 2.20223192475796944584197385901, 2.83411440003332910370433401340, 2.85255054181356782113057947793, 2.99485979106395518628171341060, 3.21187924115478014913971282982, 3.28999834391476005025175943919, 3.79002237117560914939173803414, 3.84128805810088760924826044197, 3.92480523196346363651629332528, 3.94282062098976836028335362510, 4.22467780332994002063946383692, 4.60241423104244304136854819574, 4.81136077351872981491203586629, 4.86805007066755470122665107195, 4.89820870246500834362846844918, 4.91977418794607829188825584376

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