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Properties

Label 10-4235e5-1.1-c1e5-0-1
Degree $10$
Conductor $1.362\times 10^{18}$
Sign $1$
Analytic cond. $4.42234\times 10^{7}$
Root an. cond. $5.81520$
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  + 2·2-s − 2·3-s − 4-s − 5·5-s − 4·6-s − 5·7-s − 8·8-s − 4·9-s − 10·10-s + 2·12-s + 12·13-s − 10·14-s + 10·15-s − 9·16-s + 14·17-s − 8·18-s + 9·19-s + 5·20-s + 10·21-s + 17·23-s + 16·24-s + 15·25-s + 24·26-s + 9·27-s + 5·28-s + 3·29-s + 20·30-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s − 1/2·4-s − 2.23·5-s − 1.63·6-s − 1.88·7-s − 2.82·8-s − 4/3·9-s − 3.16·10-s + 0.577·12-s + 3.32·13-s − 2.67·14-s + 2.58·15-s − 9/4·16-s + 3.39·17-s − 1.88·18-s + 2.06·19-s + 1.11·20-s + 2.18·21-s + 3.54·23-s + 3.26·24-s + 3·25-s + 4.70·26-s + 1.73·27-s + 0.944·28-s + 0.557·29-s + 3.65·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 7^{5} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 7^{5} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(5^{5} \cdot 7^{5} \cdot 11^{10}\)
Sign: $1$
Analytic conductor: \(4.42234\times 10^{7}\)
Root analytic conductor: \(5.81520\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 5^{5} \cdot 7^{5} \cdot 11^{10} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.208897069\)
\(L(\frac12)\) \(\approx\) \(5.208897069\)
\(L(\frac{3}{2})\) 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)$
bad5$C_1$ \( ( 1 + T )^{5} \)
7$C_1$ \( ( 1 + T )^{5} \)
11 \( 1 \)
good2$C_2 \wr S_5$ \( 1 - p T + 5 T^{2} - p^{2} T^{3} + 3 p T^{4} - T^{5} + 3 p^{2} T^{6} - p^{4} T^{7} + 5 p^{3} T^{8} - p^{5} T^{9} + p^{5} T^{10} \)
3$C_2 \wr S_5$ \( 1 + 2 T + 8 T^{2} + 5 p T^{3} + 28 T^{4} + 55 T^{5} + 28 p T^{6} + 5 p^{3} T^{7} + 8 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 12 T + 77 T^{2} - 23 p T^{3} + 790 T^{4} - 2147 T^{5} + 790 p T^{6} - 23 p^{3} T^{7} + 77 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 14 T + 135 T^{2} - 917 T^{3} + 5154 T^{4} - 23169 T^{5} + 5154 p T^{6} - 917 p^{2} T^{7} + 135 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 9 T + 58 T^{2} - 238 T^{3} + 1198 T^{4} - 4805 T^{5} + 1198 p T^{6} - 238 p^{2} T^{7} + 58 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 17 T + 9 p T^{2} - 77 p T^{3} + 11837 T^{4} - 63547 T^{5} + 11837 p T^{6} - 77 p^{3} T^{7} + 9 p^{4} T^{8} - 17 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 3 T + 41 T^{2} - 155 T^{3} + 1619 T^{4} - 7141 T^{5} + 1619 p T^{6} - 155 p^{2} T^{7} + 41 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 2 T + 70 T^{2} - 193 T^{3} + 3574 T^{4} - 6763 T^{5} + 3574 p T^{6} - 193 p^{2} T^{7} + 70 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 4 T + 126 T^{2} - 119 T^{3} + 5930 T^{4} + 3197 T^{5} + 5930 p T^{6} - 119 p^{2} T^{7} + 126 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 15 T + 169 T^{2} + 1461 T^{3} + 11491 T^{4} + 73989 T^{5} + 11491 p T^{6} + 1461 p^{2} T^{7} + 169 p^{3} T^{8} + 15 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 4 T + 96 T^{2} - 3 p T^{3} + 5530 T^{4} - 7273 T^{5} + 5530 p T^{6} - 3 p^{3} T^{7} + 96 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 2 T + 112 T^{2} + 116 T^{3} + 5313 T^{4} + 2319 T^{5} + 5313 p T^{6} + 116 p^{2} T^{7} + 112 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 6 T + 136 T^{2} - 533 T^{3} + 11604 T^{4} - 45703 T^{5} + 11604 p T^{6} - 533 p^{2} T^{7} + 136 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 6 T + 53 T^{2} - 95 T^{3} + 3050 T^{4} + 15029 T^{5} + 3050 p T^{6} - 95 p^{2} T^{7} + 53 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 20 T + 392 T^{2} - 4614 T^{3} + 51795 T^{4} - 415935 T^{5} + 51795 p T^{6} - 4614 p^{2} T^{7} + 392 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 3 T + 202 T^{2} - 1224 T^{3} + 18878 T^{4} - 134773 T^{5} + 18878 p T^{6} - 1224 p^{2} T^{7} + 202 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 6 T + 280 T^{2} + 1831 T^{3} + 34362 T^{4} + 199523 T^{5} + 34362 p T^{6} + 1831 p^{2} T^{7} + 280 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 11 T + 192 T^{2} - 1036 T^{3} + 170 p T^{4} - 32921 T^{5} + 170 p^{2} T^{6} - 1036 p^{2} T^{7} + 192 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 19 T + 408 T^{2} - 4296 T^{3} + 54112 T^{4} - 421467 T^{5} + 54112 p T^{6} - 4296 p^{2} T^{7} + 408 p^{3} T^{8} - 19 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 8 T + 341 T^{2} - 1949 T^{3} + 49168 T^{4} - 213589 T^{5} + 49168 p T^{6} - 1949 p^{2} T^{7} + 341 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - T + 232 T^{2} + 215 T^{3} + 25995 T^{4} + 48065 T^{5} + 25995 p T^{6} + 215 p^{2} T^{7} + 232 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 7 T + 22 T^{2} + 2241 T^{3} + 21549 T^{4} + 31355 T^{5} + 21549 p T^{6} + 2241 p^{2} T^{7} + 22 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.91407098353238139750879768728, −4.84638590837876208877110827572, −4.77899338469192790621392219256, −4.48918212069836386600141040589, −4.37386749815695823173131707372, −4.00549606720143027905888057913, −3.86316167311184998333634654642, −3.71826085695403617362708798645, −3.68024066732389953269494050081, −3.34275068482687584315840600989, −3.30972878040852916168197877276, −3.29663060391462366224144840298, −3.17256638510101664708205879239, −3.12387750033167710660828385979, −2.93262686916597054289907802575, −2.65037427453780384250904573448, −2.35711803916282965993744056785, −2.11812162826334360435177082534, −1.42899414160784797388980513927, −1.27373533064874860275732586555, −1.18424819010794324528986216767, −0.70019188310743983926598009140, −0.64539957013952332610855709758, −0.55772027239019429721422969603, −0.50608298862596667924254860014, 0.50608298862596667924254860014, 0.55772027239019429721422969603, 0.64539957013952332610855709758, 0.70019188310743983926598009140, 1.18424819010794324528986216767, 1.27373533064874860275732586555, 1.42899414160784797388980513927, 2.11812162826334360435177082534, 2.35711803916282965993744056785, 2.65037427453780384250904573448, 2.93262686916597054289907802575, 3.12387750033167710660828385979, 3.17256638510101664708205879239, 3.29663060391462366224144840298, 3.30972878040852916168197877276, 3.34275068482687584315840600989, 3.68024066732389953269494050081, 3.71826085695403617362708798645, 3.86316167311184998333634654642, 4.00549606720143027905888057913, 4.37386749815695823173131707372, 4.48918212069836386600141040589, 4.77899338469192790621392219256, 4.84638590837876208877110827572, 4.91407098353238139750879768728

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