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Properties

Label 10-5610e5-1.1-c1e5-0-2
Degree $10$
Conductor $5.557\times 10^{18}$
Sign $1$
Analytic cond. $1.80384\times 10^{8}$
Root an. cond. $6.69298$
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 + 5·5-s + 25·6-s + 7-s + 35·8-s + 15·9-s + 25·10-s + 5·11-s + 75·12-s + 9·13-s + 5·14-s + 25·15-s + 70·16-s + 5·17-s + 75·18-s − 5·19-s + 75·20-s + 5·21-s + 25·22-s + 11·23-s + 175·24-s + 15·25-s + 45·26-s + 35·27-s + 15·28-s + ⋯
L(s)  = 1  + 3.53·2-s + 2.88·3-s + 15/2·4-s + 2.23·5-s + 10.2·6-s + 0.377·7-s + 12.3·8-s + 5·9-s + 7.90·10-s + 1.50·11-s + 21.6·12-s + 2.49·13-s + 1.33·14-s + 6.45·15-s + 35/2·16-s + 1.21·17-s + 17.6·18-s − 1.14·19-s + 16.7·20-s + 1.09·21-s + 5.33·22-s + 2.29·23-s + 35.7·24-s + 3·25-s + 8.82·26-s + 6.73·27-s + 2.83·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{5} \cdot 5^{5} \cdot 11^{5} \cdot 17^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{5} \cdot 5^{5} \cdot 11^{5} \cdot 17^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{5} \cdot 3^{5} \cdot 5^{5} \cdot 11^{5} \cdot 17^{5}\)
Sign: $1$
Analytic conductor: \(1.80384\times 10^{8}\)
Root analytic conductor: \(6.69298\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{5} \cdot 3^{5} \cdot 5^{5} \cdot 11^{5} \cdot 17^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2628.891462\)
\(L(\frac12)\) \(\approx\) \(2628.891462\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{5} \)
5$C_1$ \( ( 1 - T )^{5} \)
11$C_1$ \( ( 1 - T )^{5} \)
17$C_1$ \( ( 1 - T )^{5} \)
good7$C_2 \wr S_5$ \( 1 - T + 11 T^{2} + 4 T^{3} + 50 T^{4} + 90 T^{5} + 50 p T^{6} + 4 p^{2} T^{7} + 11 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 9 T + 73 T^{2} - 412 T^{3} + 1954 T^{4} - 7686 T^{5} + 1954 p T^{6} - 412 p^{2} T^{7} + 73 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 5 T + 31 T^{2} + 124 T^{3} + 602 T^{4} + 3342 T^{5} + 602 p T^{6} + 124 p^{2} T^{7} + 31 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 11 T + 59 T^{2} - 308 T^{3} + 2258 T^{4} - 13346 T^{5} + 2258 p T^{6} - 308 p^{2} T^{7} + 59 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 2 T + 25 T^{2} - 72 T^{3} - 30 T^{4} - 7380 T^{5} - 30 p T^{6} - 72 p^{2} T^{7} + 25 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 7 T + 131 T^{2} + 772 T^{3} + 7506 T^{4} + 34666 T^{5} + 7506 p T^{6} + 772 p^{2} T^{7} + 131 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 23 T + 353 T^{2} - 3716 T^{3} + 31458 T^{4} - 209498 T^{5} + 31458 p T^{6} - 3716 p^{2} T^{7} + 353 p^{3} T^{8} - 23 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 14 T + 133 T^{2} - 1480 T^{3} + 11682 T^{4} - 71444 T^{5} + 11682 p T^{6} - 1480 p^{2} T^{7} + 133 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 8 T + 183 T^{2} + 1184 T^{3} + 14426 T^{4} + 72752 T^{5} + 14426 p T^{6} + 1184 p^{2} T^{7} + 183 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 6 T + 75 T^{2} - 40 T^{3} + 810 T^{4} + 19164 T^{5} + 810 p T^{6} - 40 p^{2} T^{7} + 75 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 6 T + 209 T^{2} + 1064 T^{3} + 19842 T^{4} + 80964 T^{5} + 19842 p T^{6} + 1064 p^{2} T^{7} + 209 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 20 T + 327 T^{2} - 3824 T^{3} + 39514 T^{4} - 320952 T^{5} + 39514 p T^{6} - 3824 p^{2} T^{7} + 327 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 5 T + 9 T^{2} - 436 T^{3} + 1298 T^{4} + 11246 T^{5} + 1298 p T^{6} - 436 p^{2} T^{7} + 9 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 3 T + 231 T^{2} - 660 T^{3} + 24434 T^{4} - 61378 T^{5} + 24434 p T^{6} - 660 p^{2} T^{7} + 231 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 2 T + 99 T^{2} + 952 T^{3} + 8874 T^{4} + 75724 T^{5} + 8874 p T^{6} + 952 p^{2} T^{7} + 99 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 12 T + 309 T^{2} - 3472 T^{3} + 41170 T^{4} - 378952 T^{5} + 41170 p T^{6} - 3472 p^{2} T^{7} + 309 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 14 T + 203 T^{2} - 2248 T^{3} + 23626 T^{4} - 245076 T^{5} + 23626 p T^{6} - 2248 p^{2} T^{7} + 203 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 17 T + 351 T^{2} - 3308 T^{3} + 42970 T^{4} - 315158 T^{5} + 42970 p T^{6} - 3308 p^{2} T^{7} + 351 p^{3} T^{8} - 17 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 6 T + 229 T^{2} + 1672 T^{3} + 26738 T^{4} + 212004 T^{5} + 26738 p T^{6} + 1672 p^{2} T^{7} + 229 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 23 T + 613 T^{2} - 8372 T^{3} + 124474 T^{4} - 1175786 T^{5} + 124474 p T^{6} - 8372 p^{2} T^{7} + 613 p^{3} T^{8} - 23 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.60744040096848757850966224048, −4.52618933118405881837365612986, −4.40749614557740349131429485287, −4.23605849450824056652983535233, −4.22722395619373937747667991679, −3.84665533452265178718053498890, −3.81290244950582989460129417386, −3.71353187698148540983936161841, −3.56315637138432770117014145080, −3.36437885708685805958808398119, −3.17523543668281190622148519576, −3.06511736749740562871737382429, −2.93408212814569450022812408960, −2.70986705647856060074943350634, −2.69766670758317421724021023760, −2.14333421439338145175656596766, −2.12791798608328171588820118731, −2.08832732632262895135142922351, −2.03787053862672725729994354366, −1.92234721131880709079098053999, −1.22379018860731284561334829791, −1.18574482660672988826047216796, −1.13789753752006385738872059112, −1.05076889562059179320160805672, −0.947283994610199291711553297352, 0.947283994610199291711553297352, 1.05076889562059179320160805672, 1.13789753752006385738872059112, 1.18574482660672988826047216796, 1.22379018860731284561334829791, 1.92234721131880709079098053999, 2.03787053862672725729994354366, 2.08832732632262895135142922351, 2.12791798608328171588820118731, 2.14333421439338145175656596766, 2.69766670758317421724021023760, 2.70986705647856060074943350634, 2.93408212814569450022812408960, 3.06511736749740562871737382429, 3.17523543668281190622148519576, 3.36437885708685805958808398119, 3.56315637138432770117014145080, 3.71353187698148540983936161841, 3.81290244950582989460129417386, 3.84665533452265178718053498890, 4.22722395619373937747667991679, 4.23605849450824056652983535233, 4.40749614557740349131429485287, 4.52618933118405881837365612986, 4.60744040096848757850966224048

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