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Properties

Label 10-3720e5-1.1-c1e5-0-1
Degree $10$
Conductor $7.124\times 10^{17}$
Sign $1$
Analytic cond. $2.31259\times 10^{7}$
Root an. cond. $5.45016$
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 − 5·5-s + 3·7-s + 15·9-s + 5·11-s + 2·13-s − 25·15-s + 6·17-s + 9·19-s + 15·21-s − 3·23-s + 15·25-s + 35·27-s + 2·29-s + 5·31-s + 25·33-s − 15·35-s + 4·37-s + 10·39-s + 8·41-s + 7·43-s − 75·45-s − 2·47-s − 6·49-s + 30·51-s + 5·53-s − 25·55-s + ⋯
L(s)  = 1  + 2.88·3-s − 2.23·5-s + 1.13·7-s + 5·9-s + 1.50·11-s + 0.554·13-s − 6.45·15-s + 1.45·17-s + 2.06·19-s + 3.27·21-s − 0.625·23-s + 3·25-s + 6.73·27-s + 0.371·29-s + 0.898·31-s + 4.35·33-s − 2.53·35-s + 0.657·37-s + 1.60·39-s + 1.24·41-s + 1.06·43-s − 11.1·45-s − 0.291·47-s − 6/7·49-s + 4.20·51-s + 0.686·53-s − 3.37·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{5} \cdot 5^{5} \cdot 31^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{5} \cdot 5^{5} \cdot 31^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{15} \cdot 3^{5} \cdot 5^{5} \cdot 31^{5}\)
Sign: $1$
Analytic conductor: \(2.31259\times 10^{7}\)
Root analytic conductor: \(5.45016\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{15} \cdot 3^{5} \cdot 5^{5} \cdot 31^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(61.05361834\)
\(L(\frac12)\) \(\approx\) \(61.05361834\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{5} \)
5$C_1$ \( ( 1 + T )^{5} \)
31$C_1$ \( ( 1 - T )^{5} \)
good7$C_2 \wr S_5$ \( 1 - 3 T + 15 T^{2} - 4 p T^{3} + 170 T^{4} - 354 T^{5} + 170 p T^{6} - 4 p^{3} T^{7} + 15 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 5 T + 29 T^{2} - 148 T^{3} + 576 T^{4} - 1918 T^{5} + 576 p T^{6} - 148 p^{2} T^{7} + 29 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 2 T + 21 T^{2} + 44 T^{3} + 22 T^{4} + 1548 T^{5} + 22 p T^{6} + 44 p^{2} T^{7} + 21 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 6 T + 57 T^{2} - 240 T^{3} + 1558 T^{4} - 5204 T^{5} + 1558 p T^{6} - 240 p^{2} T^{7} + 57 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 9 T + 91 T^{2} - 540 T^{3} + 3350 T^{4} - 14534 T^{5} + 3350 p T^{6} - 540 p^{2} T^{7} + 91 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 3 T + 57 T^{2} + 196 T^{3} + 1768 T^{4} + 5554 T^{5} + 1768 p T^{6} + 196 p^{2} T^{7} + 57 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 2 T + 69 T^{2} - 124 T^{3} + 2838 T^{4} - 5876 T^{5} + 2838 p T^{6} - 124 p^{2} T^{7} + 69 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 4 T + 117 T^{2} - 356 T^{3} + 6086 T^{4} - 15520 T^{5} + 6086 p T^{6} - 356 p^{2} T^{7} + 117 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 8 T + 133 T^{2} - 704 T^{3} + 7074 T^{4} - 31344 T^{5} + 7074 p T^{6} - 704 p^{2} T^{7} + 133 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 7 T + 155 T^{2} - 804 T^{3} + 11550 T^{4} - 48506 T^{5} + 11550 p T^{6} - 804 p^{2} T^{7} + 155 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 2 T + 203 T^{2} + 312 T^{3} + 17674 T^{4} + 20620 T^{5} + 17674 p T^{6} + 312 p^{2} T^{7} + 203 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 5 T + 73 T^{2} - 524 T^{3} + 5722 T^{4} - 14638 T^{5} + 5722 p T^{6} - 524 p^{2} T^{7} + 73 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 6 T + 271 T^{2} - 1228 T^{3} + 30330 T^{4} - 103196 T^{5} + 30330 p T^{6} - 1228 p^{2} T^{7} + 271 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 16 T + 361 T^{2} - 3808 T^{3} + 46898 T^{4} - 345248 T^{5} + 46898 p T^{6} - 3808 p^{2} T^{7} + 361 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 22 T + 211 T^{2} - 1084 T^{3} + 6934 T^{4} - 60924 T^{5} + 6934 p T^{6} - 1084 p^{2} T^{7} + 211 p^{3} T^{8} - 22 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 9 T + 159 T^{2} + 1272 T^{3} + 15634 T^{4} + 108862 T^{5} + 15634 p T^{6} + 1272 p^{2} T^{7} + 159 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 9 T + 275 T^{2} - 2584 T^{3} + 34848 T^{4} - 279390 T^{5} + 34848 p T^{6} - 2584 p^{2} T^{7} + 275 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 15 T + 175 T^{2} - 1372 T^{3} + 12430 T^{4} - 76202 T^{5} + 12430 p T^{6} - 1372 p^{2} T^{7} + 175 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 8 T + 279 T^{2} + 2064 T^{3} + 36050 T^{4} + 236496 T^{5} + 36050 p T^{6} + 2064 p^{2} T^{7} + 279 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 17 T + 365 T^{2} - 3384 T^{3} + 45454 T^{4} - 325222 T^{5} + 45454 p T^{6} - 3384 p^{2} T^{7} + 365 p^{3} T^{8} - 17 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 18 T + 461 T^{2} - 6328 T^{3} + 87186 T^{4} - 890220 T^{5} + 87186 p T^{6} - 6328 p^{2} T^{7} + 461 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.88807300046132645981146767825, −4.59803215270567025949538803733, −4.58405995509532895966011109131, −4.45544771438207239969943564905, −4.34627763046871287285296227783, −4.20749040002192000155008074388, −3.99778191203009782755213190250, −3.71577321708172761137896204276, −3.61659356244709297057449253473, −3.61293583086676560652927446779, −3.20046448552534301009969322038, −3.17996593809024746484793246303, −3.17443161230748604868076008912, −3.11141233250029176863144926494, −2.76151941451924083031857911591, −2.17711445951069111187383185610, −2.17489107968255585438146228999, −2.16388069779699228386709754003, −1.97948661865930659740241934556, −1.71226668139227306190439373060, −1.20915762489065957316541098815, −0.954605222188967690795453244787, −0.917690129591161090674844710887, −0.74400980576831977973382701753, −0.71479527836383060864649423215, 0.71479527836383060864649423215, 0.74400980576831977973382701753, 0.917690129591161090674844710887, 0.954605222188967690795453244787, 1.20915762489065957316541098815, 1.71226668139227306190439373060, 1.97948661865930659740241934556, 2.16388069779699228386709754003, 2.17489107968255585438146228999, 2.17711445951069111187383185610, 2.76151941451924083031857911591, 3.11141233250029176863144926494, 3.17443161230748604868076008912, 3.17996593809024746484793246303, 3.20046448552534301009969322038, 3.61293583086676560652927446779, 3.61659356244709297057449253473, 3.71577321708172761137896204276, 3.99778191203009782755213190250, 4.20749040002192000155008074388, 4.34627763046871287285296227783, 4.45544771438207239969943564905, 4.58405995509532895966011109131, 4.59803215270567025949538803733, 4.88807300046132645981146767825

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