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Properties

Label 12-2520e6-1.1-c1e6-0-7
Degree $12$
Conductor $2.561\times 10^{20}$
Sign $1$
Analytic cond. $6.63843\times 10^{7}$
Root an. cond. $4.48578$
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  − 3·5-s − 3·11-s − 6·13-s + 6·17-s − 3·19-s − 3·23-s + 3·25-s − 12·29-s − 12·31-s − 3·37-s + 18·41-s + 3·47-s + 6·49-s − 15·53-s + 9·55-s + 30·59-s + 18·65-s + 24·71-s − 18·73-s − 6·79-s − 12·83-s − 18·85-s + 9·95-s + 48·97-s + 18·101-s − 12·103-s − 12·109-s + ⋯
L(s)  = 1  − 1.34·5-s − 0.904·11-s − 1.66·13-s + 1.45·17-s − 0.688·19-s − 0.625·23-s + 3/5·25-s − 2.22·29-s − 2.15·31-s − 0.493·37-s + 2.81·41-s + 0.437·47-s + 6/7·49-s − 2.06·53-s + 1.21·55-s + 3.90·59-s + 2.23·65-s + 2.84·71-s − 2.10·73-s − 0.675·79-s − 1.31·83-s − 1.95·85-s + 0.923·95-s + 4.87·97-s + 1.79·101-s − 1.18·103-s − 1.14·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(6.63843\times 10^{7}\)
Root analytic conductor: \(4.48578\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.398225393\)
\(L(\frac12)\) \(\approx\) \(1.398225393\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( ( 1 + T + T^{2} )^{3} \)
7 \( 1 - 6 T^{2} - 4 T^{3} - 6 p T^{4} + p^{3} T^{6} \)
good11 \( 1 + 3 T + 12 T^{2} + 111 T^{3} + 222 T^{4} + 741 T^{5} + 5074 T^{6} + 741 p T^{7} + 222 p^{2} T^{8} + 111 p^{3} T^{9} + 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 + 3 T + 24 T^{2} + 81 T^{3} + 24 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 - 6 T + 39 T^{2} - 214 T^{3} + 756 T^{4} - 2694 T^{5} + 12725 T^{6} - 2694 p T^{7} + 756 p^{2} T^{8} - 214 p^{3} T^{9} + 39 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
19 \( ( 1 - 7 T + p T^{2} )^{3}( 1 + 8 T + p T^{2} )^{3} \)
23 \( 1 + 3 T - 36 T^{2} - 97 T^{3} + 642 T^{4} + 597 T^{5} - 13546 T^{6} + 597 p T^{7} + 642 p^{2} T^{8} - 97 p^{3} T^{9} - 36 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 + 6 T + 81 T^{2} + 300 T^{3} + 81 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 + 12 T + 78 T^{2} + 204 T^{3} - 1218 T^{4} - 18168 T^{5} - 127402 T^{6} - 18168 p T^{7} - 1218 p^{2} T^{8} + 204 p^{3} T^{9} + 78 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 3 T - 15 T^{2} + 390 T^{3} + 165 T^{4} - 5037 T^{5} + 92954 T^{6} - 5037 p T^{7} + 165 p^{2} T^{8} + 390 p^{3} T^{9} - 15 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 - 9 T + 3 p T^{2} - 680 T^{3} + 3 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 + 90 T^{2} + 88 T^{3} + 90 p T^{4} + p^{3} T^{6} )^{2} \)
47 \( 1 - 3 T - 60 T^{2} + 637 T^{3} + 252 T^{4} - 16287 T^{5} + 142082 T^{6} - 16287 p T^{7} + 252 p^{2} T^{8} + 637 p^{3} T^{9} - 60 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 15 T + 6 T^{2} - 7 T^{3} + 11664 T^{4} + 54399 T^{5} - 166888 T^{6} + 54399 p T^{7} + 11664 p^{2} T^{8} - 7 p^{3} T^{9} + 6 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 30 T + 441 T^{2} - 5010 T^{3} + 52200 T^{4} - 489990 T^{5} + 4032079 T^{6} - 489990 p T^{7} + 52200 p^{2} T^{8} - 5010 p^{3} T^{9} + 441 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \)
61 \( ( 1 - p T^{2} + p^{2} T^{4} )^{3} \)
67 \( 1 - 162 T^{2} + 176 T^{3} + 15390 T^{4} - 14256 T^{5} - 1149078 T^{6} - 14256 p T^{7} + 15390 p^{2} T^{8} + 176 p^{3} T^{9} - 162 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 - 12 T + 27 T^{2} + 396 T^{3} + 27 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 - 3 T - 36 T^{2} + 887 T^{3} - 36 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )( 1 + 21 T + 123 T^{2} + 266 T^{3} + 123 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} ) \)
79 \( 1 + 6 T - 114 T^{2} - 932 T^{3} + 5154 T^{4} + 32646 T^{5} - 177042 T^{6} + 32646 p T^{7} + 5154 p^{2} T^{8} - 932 p^{3} T^{9} - 114 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 + 6 T + 243 T^{2} + 948 T^{3} + 243 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 177 T^{2} - 584 T^{3} + 15576 T^{4} + 51684 T^{5} - 1293079 T^{6} + 51684 p T^{7} + 15576 p^{2} T^{8} - 584 p^{3} T^{9} - 177 p^{4} T^{10} + p^{6} T^{12} \)
97 \( ( 1 - 8 T + p T^{2} )^{6} \)
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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

−4.64154689285952832879397964412, −4.50863391398788622023391962828, −4.19323749898078052748218584901, −4.15938734662628161366332972516, −3.97403349787429082099633749470, −3.76904693673717116943131986337, −3.72685031315120295375413500656, −3.69794625239919481579365373628, −3.63014441810407294012281518870, −3.20720005963710417444747089498, −3.16298025702378122272977541460, −2.86297817930385685429448960310, −2.78080561889435727068150188087, −2.48336193293474823352701919832, −2.36934357008553343211191694450, −2.33743245403336859295163607268, −2.26739233033539666792820374792, −1.74580516398046506730680512033, −1.61796169415450801530094384873, −1.58334214081825357423969511893, −1.40785434828743566222938673644, −0.71027180924549681782287393462, −0.57845352719885983510987331839, −0.57477994180979340253163553837, −0.20306336867386223694862182147, 0.20306336867386223694862182147, 0.57477994180979340253163553837, 0.57845352719885983510987331839, 0.71027180924549681782287393462, 1.40785434828743566222938673644, 1.58334214081825357423969511893, 1.61796169415450801530094384873, 1.74580516398046506730680512033, 2.26739233033539666792820374792, 2.33743245403336859295163607268, 2.36934357008553343211191694450, 2.48336193293474823352701919832, 2.78080561889435727068150188087, 2.86297817930385685429448960310, 3.16298025702378122272977541460, 3.20720005963710417444747089498, 3.63014441810407294012281518870, 3.69794625239919481579365373628, 3.72685031315120295375413500656, 3.76904693673717116943131986337, 3.97403349787429082099633749470, 4.15938734662628161366332972516, 4.19323749898078052748218584901, 4.50863391398788622023391962828, 4.64154689285952832879397964412

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

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