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Properties

Label 10-6012e5-1.1-c1e5-0-0
Degree $10$
Conductor $7.854\times 10^{18}$
Sign $1$
Analytic cond. $2.54964\times 10^{8}$
Root an. cond. $6.92864$
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  − 10·5-s + 9·7-s − 5·11-s − 4·13-s + 2·17-s + 5·19-s − 6·23-s + 35·25-s + 5·29-s + 9·31-s − 90·35-s + 8·37-s + 4·41-s + 8·43-s − 13·47-s + 30·49-s + 2·53-s + 50·55-s − 4·59-s + 11·61-s + 40·65-s + 28·67-s − 2·71-s + 8·73-s − 45·77-s − 10·79-s − 2·83-s + ⋯
L(s)  = 1  − 4.47·5-s + 3.40·7-s − 1.50·11-s − 1.10·13-s + 0.485·17-s + 1.14·19-s − 1.25·23-s + 7·25-s + 0.928·29-s + 1.61·31-s − 15.2·35-s + 1.31·37-s + 0.624·41-s + 1.21·43-s − 1.89·47-s + 30/7·49-s + 0.274·53-s + 6.74·55-s − 0.520·59-s + 1.40·61-s + 4.96·65-s + 3.42·67-s − 0.237·71-s + 0.936·73-s − 5.12·77-s − 1.12·79-s − 0.219·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 167^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 167^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{10} \cdot 3^{10} \cdot 167^{5}\)
Sign: $1$
Analytic conductor: \(2.54964\times 10^{8}\)
Root analytic conductor: \(6.92864\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{10} \cdot 3^{10} \cdot 167^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.055660749\)
\(L(\frac12)\) \(\approx\) \(4.055660749\)
\(L(\frac{3}{2})\) 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 \( 1 \)
167$C_1$ \( ( 1 - T )^{5} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{5} \)
7$C_2 \wr S_5$ \( 1 - 9 T + 51 T^{2} - 223 T^{3} + 787 T^{4} - 2265 T^{5} + 787 p T^{6} - 223 p^{2} T^{7} + 51 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 5 T + 43 T^{2} + 191 T^{3} + 849 T^{4} + 3017 T^{5} + 849 p T^{6} + 191 p^{2} T^{7} + 43 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 4 T + 45 T^{2} + 160 T^{3} + 1006 T^{4} + 2840 T^{5} + 1006 p T^{6} + 160 p^{2} T^{7} + 45 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 2 T + 25 T^{2} - 96 T^{3} + 662 T^{4} - 1308 T^{5} + 662 p T^{6} - 96 p^{2} T^{7} + 25 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 5 T + 53 T^{2} - 279 T^{3} + 1821 T^{4} - 6547 T^{5} + 1821 p T^{6} - 279 p^{2} T^{7} + 53 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 6 T + 63 T^{2} + 192 T^{3} + 1414 T^{4} + 2452 T^{5} + 1414 p T^{6} + 192 p^{2} T^{7} + 63 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 5 T + 117 T^{2} - 541 T^{3} + 6005 T^{4} - 22981 T^{5} + 6005 p T^{6} - 541 p^{2} T^{7} + 117 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 9 T + 119 T^{2} - 909 T^{3} + 6257 T^{4} - 39425 T^{5} + 6257 p T^{6} - 909 p^{2} T^{7} + 119 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 8 T + 141 T^{2} - 1104 T^{3} + 8998 T^{4} - 60080 T^{5} + 8998 p T^{6} - 1104 p^{2} T^{7} + 141 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 4 T + 109 T^{2} - 424 T^{3} + 6474 T^{4} - 20392 T^{5} + 6474 p T^{6} - 424 p^{2} T^{7} + 109 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 8 T + 135 T^{2} - 992 T^{3} + 9706 T^{4} - 56752 T^{5} + 9706 p T^{6} - 992 p^{2} T^{7} + 135 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 13 T + 261 T^{2} + 2283 T^{3} + 539 p T^{4} + 157087 T^{5} + 539 p^{2} T^{6} + 2283 p^{2} T^{7} + 261 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 2 T + 141 T^{2} - 216 T^{3} + 11910 T^{4} - 17068 T^{5} + 11910 p T^{6} - 216 p^{2} T^{7} + 141 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 4 T + p T^{2} + 120 T^{3} + 4414 T^{4} + 36008 T^{5} + 4414 p T^{6} + 120 p^{2} T^{7} + p^{4} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 11 T + 229 T^{2} - 2491 T^{3} + 24489 T^{4} - 221163 T^{5} + 24489 p T^{6} - 2491 p^{2} T^{7} + 229 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 28 T + 491 T^{2} - 5616 T^{3} + 53302 T^{4} - 436232 T^{5} + 53302 p T^{6} - 5616 p^{2} T^{7} + 491 p^{3} T^{8} - 28 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 2 T + 175 T^{2} + 320 T^{3} + 17542 T^{4} + 36060 T^{5} + 17542 p T^{6} + 320 p^{2} T^{7} + 175 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 8 T + 241 T^{2} - 616 T^{3} + 19206 T^{4} + 4320 T^{5} + 19206 p T^{6} - 616 p^{2} T^{7} + 241 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 10 T + 139 T^{2} + 1552 T^{3} + 15962 T^{4} + 195500 T^{5} + 15962 p T^{6} + 1552 p^{2} T^{7} + 139 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 2 T + 179 T^{2} + 656 T^{3} + 20622 T^{4} + 69980 T^{5} + 20622 p T^{6} + 656 p^{2} T^{7} + 179 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 17 T + 485 T^{2} - 5689 T^{3} + 89097 T^{4} - 743149 T^{5} + 89097 p T^{6} - 5689 p^{2} T^{7} + 485 p^{3} T^{8} - 17 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 27 T + 521 T^{2} + 8059 T^{3} + 104425 T^{4} + 1093259 T^{5} + 104425 p T^{6} + 8059 p^{2} T^{7} + 521 p^{3} T^{8} + 27 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.58336815585772546741611115094, −4.45527393144613737258945312346, −4.45503180300309749304099542893, −4.36207095646761771736990946529, −4.33653068101832150671045387553, −3.94028393600128410470057689228, −3.79730173275278646819600034149, −3.64165772659627140897306247006, −3.62468217508920656778470352265, −3.60531847394915253201406939915, −2.98856155712789124660182847044, −2.96697834252916844500366018324, −2.96358500094648538473670534937, −2.64466689812934548514119343424, −2.53887814660390827700433733198, −2.17174007771652196462064891864, −1.87299525056923771493175416382, −1.86246907101348555792878078597, −1.82388446491274241274680364875, −1.50660874484110468289013334227, −1.01584197122131575960536029100, −0.75457279993641746987487924232, −0.58624321435360987084695345712, −0.50084900444925075292918399694, −0.38301276128921276321183727139, 0.38301276128921276321183727139, 0.50084900444925075292918399694, 0.58624321435360987084695345712, 0.75457279993641746987487924232, 1.01584197122131575960536029100, 1.50660874484110468289013334227, 1.82388446491274241274680364875, 1.86246907101348555792878078597, 1.87299525056923771493175416382, 2.17174007771652196462064891864, 2.53887814660390827700433733198, 2.64466689812934548514119343424, 2.96358500094648538473670534937, 2.96697834252916844500366018324, 2.98856155712789124660182847044, 3.60531847394915253201406939915, 3.62468217508920656778470352265, 3.64165772659627140897306247006, 3.79730173275278646819600034149, 3.94028393600128410470057689228, 4.33653068101832150671045387553, 4.36207095646761771736990946529, 4.45503180300309749304099542893, 4.45527393144613737258945312346, 4.58336815585772546741611115094

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