Properties

Label 12-5070e6-1.1-c1e6-0-3
Degree $12$
Conductor $1.698\times 10^{22}$
Sign $1$
Analytic cond. $4.40261\times 10^{9}$
Root an. cond. $6.36271$
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  − 6·3-s − 3·4-s + 21·9-s + 18·12-s + 6·16-s − 6·17-s + 10·23-s − 3·25-s − 56·27-s + 24·29-s − 63·36-s + 10·43-s − 36·48-s + 28·49-s + 36·51-s + 6·53-s − 34·61-s − 10·64-s + 18·68-s − 60·69-s + 18·75-s − 14·79-s + 126·81-s − 144·87-s − 30·92-s + 9·100-s + 20·101-s + ⋯
L(s)  = 1  − 3.46·3-s − 3/2·4-s + 7·9-s + 5.19·12-s + 3/2·16-s − 1.45·17-s + 2.08·23-s − 3/5·25-s − 10.7·27-s + 4.45·29-s − 10.5·36-s + 1.52·43-s − 5.19·48-s + 4·49-s + 5.04·51-s + 0.824·53-s − 4.35·61-s − 5/4·64-s + 2.18·68-s − 7.22·69-s + 2.07·75-s − 1.57·79-s + 14·81-s − 15.4·87-s − 3.12·92-s + 9/10·100-s + 1.99·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(4.40261\times 10^{9}\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5070} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3719471847\)
\(L(\frac12)\) \(\approx\) \(0.3719471847\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{3} \)
3 \( ( 1 + T )^{6} \)
5 \( ( 1 + T^{2} )^{3} \)
13 \( 1 \)
good7 \( 1 - 4 p T^{2} + 8 p^{2} T^{4} - 69 p^{2} T^{6} + 8 p^{4} T^{8} - 4 p^{5} T^{10} + p^{6} T^{12} \)
11 \( 1 + 4 T^{2} + 156 T^{4} + 1219 T^{6} + 156 p^{2} T^{8} + 4 p^{4} T^{10} + p^{6} T^{12} \)
17 \( ( 1 + 3 T + 47 T^{2} + 89 T^{3} + 47 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( 1 - 9 T^{2} + 1061 T^{4} - 6329 T^{6} + 1061 p^{2} T^{8} - 9 p^{4} T^{10} + p^{6} T^{12} \)
23 \( ( 1 - 5 T - 9 T^{2} + 189 T^{3} - 9 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( ( 1 - 12 T + 72 T^{2} - 11 p T^{3} + 72 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 - 60 T^{2} + 212 T^{4} + 42643 T^{6} + 212 p^{2} T^{8} - 60 p^{4} T^{10} + p^{6} T^{12} \)
37 \( 1 - 156 T^{2} + 11960 T^{4} - 554177 T^{6} + 11960 p^{2} T^{8} - 156 p^{4} T^{10} + p^{6} T^{12} \)
41 \( 1 - 73 T^{2} + 461 T^{4} + 89103 T^{6} + 461 p^{2} T^{8} - 73 p^{4} T^{10} + p^{6} T^{12} \)
43 \( ( 1 - 5 T + 121 T^{2} - 389 T^{3} + 121 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 79 T^{2} + 606 T^{4} + 125621 T^{6} + 606 p^{2} T^{8} - 79 p^{4} T^{10} + p^{6} T^{12} \)
53 \( ( 1 - 3 T + 141 T^{2} - 305 T^{3} + 141 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 165 T^{2} + 11237 T^{4} - 579713 T^{6} + 11237 p^{2} T^{8} - 165 p^{4} T^{10} + p^{6} T^{12} \)
61 \( ( 1 + 17 T + 179 T^{2} + 1473 T^{3} + 179 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 389 T^{2} + 63877 T^{4} - 5668601 T^{6} + 63877 p^{2} T^{8} - 389 p^{4} T^{10} + p^{6} T^{12} \)
71 \( 1 - 197 T^{2} + 11817 T^{4} - 407513 T^{6} + 11817 p^{2} T^{8} - 197 p^{4} T^{10} + p^{6} T^{12} \)
73 \( 1 - 153 T^{2} + 13157 T^{4} - 815273 T^{6} + 13157 p^{2} T^{8} - 153 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 + 7 T + 181 T^{2} + 1015 T^{3} + 181 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 268 T^{2} + 42912 T^{4} - 4229893 T^{6} + 42912 p^{2} T^{8} - 268 p^{4} T^{10} + p^{6} T^{12} \)
89 \( 1 - 349 T^{2} + 62457 T^{4} - 6912577 T^{6} + 62457 p^{2} T^{8} - 349 p^{4} T^{10} + p^{6} T^{12} \)
97 \( 1 - 540 T^{2} + 125280 T^{4} - 15967897 T^{6} + 125280 p^{2} T^{8} - 540 p^{4} T^{10} + p^{6} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.31049750589098281216614589243, −4.26540727912016278546452516353, −4.18217783411873223872517751340, −3.90860469256281205758604388599, −3.81180721996032855076429499071, −3.73408724790913424148030118897, −3.32869962824674179506218151530, −3.29711475573471965272622547439, −3.08865171667360501394372639896, −2.93487918029643805934258069774, −2.70181368618721728365467700752, −2.67124140197049916935529837607, −2.61244253344135603191622391879, −2.24504212541803203646051151749, −2.02759929855120867056044307394, −1.98485520790170391133114084274, −1.64974660127367491576258836081, −1.47793356669188518486648387051, −1.12064687734971819372143746988, −1.03768067403991475902943316255, −1.00434870795215488568264849062, −0.865356396450850205167519087393, −0.67699168484727859841905461558, −0.30340717774835447534491289212, −0.15617528345317018793389955649, 0.15617528345317018793389955649, 0.30340717774835447534491289212, 0.67699168484727859841905461558, 0.865356396450850205167519087393, 1.00434870795215488568264849062, 1.03768067403991475902943316255, 1.12064687734971819372143746988, 1.47793356669188518486648387051, 1.64974660127367491576258836081, 1.98485520790170391133114084274, 2.02759929855120867056044307394, 2.24504212541803203646051151749, 2.61244253344135603191622391879, 2.67124140197049916935529837607, 2.70181368618721728365467700752, 2.93487918029643805934258069774, 3.08865171667360501394372639896, 3.29711475573471965272622547439, 3.32869962824674179506218151530, 3.73408724790913424148030118897, 3.81180721996032855076429499071, 3.90860469256281205758604388599, 4.18217783411873223872517751340, 4.26540727912016278546452516353, 4.31049750589098281216614589243

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

Plot not available for L-functions of degree greater than 10.