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Properties

Label 12-2646e6-1.1-c1e6-0-0
Degree $12$
Conductor $3.432\times 10^{20}$
Sign $1$
Analytic cond. $8.89614\times 10^{7}$
Root an. cond. $4.59656$
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  − 3·2-s + 3·4-s + 5-s + 2·8-s − 3·10-s + 11-s − 8·13-s − 9·16-s + 8·17-s − 6·19-s + 3·20-s − 3·22-s + 7·23-s + 9·25-s + 24·26-s + 5·29-s − 20·31-s + 9·32-s − 24·34-s − 6·37-s + 18·38-s + 2·40-s − 6·43-s + 3·44-s − 21·46-s − 9·47-s − 27·50-s + ⋯
L(s)  = 1  − 2.12·2-s + 3/2·4-s + 0.447·5-s + 0.707·8-s − 0.948·10-s + 0.301·11-s − 2.21·13-s − 9/4·16-s + 1.94·17-s − 1.37·19-s + 0.670·20-s − 0.639·22-s + 1.45·23-s + 9/5·25-s + 4.70·26-s + 0.928·29-s − 3.59·31-s + 1.59·32-s − 4.11·34-s − 0.986·37-s + 2.91·38-s + 0.316·40-s − 0.914·43-s + 0.452·44-s − 3.09·46-s − 1.31·47-s − 3.81·50-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{18} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(8.89614\times 10^{7}\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{18} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.4156409291\)
\(L(\frac12)\) \(\approx\) \(0.4156409291\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T + T^{2} )^{3} \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - T - 8 T^{2} + 17 T^{3} + 23 T^{4} - 52 T^{5} - 11 T^{6} - 52 p T^{7} + 23 p^{2} T^{8} + 17 p^{3} T^{9} - 8 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - T - 26 T^{2} + 23 T^{3} + 37 p T^{4} - 202 T^{5} - 4853 T^{6} - 202 p T^{7} + 37 p^{3} T^{8} + 23 p^{3} T^{9} - 26 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 8 T + 24 T^{2} + 42 T^{3} - 32 T^{4} - 1408 T^{5} - 7901 T^{6} - 1408 p T^{7} - 32 p^{2} T^{8} + 42 p^{3} T^{9} + 24 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
17 \( ( 1 - 4 T + 39 T^{2} - 112 T^{3} + 39 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( ( 1 + 3 T + 21 T^{2} + 65 T^{3} + 21 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 7 T - 32 T^{2} + 83 T^{3} + 2423 T^{4} - 3946 T^{5} - 46865 T^{6} - 3946 p T^{7} + 2423 p^{2} T^{8} + 83 p^{3} T^{9} - 32 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 5 T + 4 T^{2} - 251 T^{3} + 197 T^{4} + 3418 T^{5} + 20293 T^{6} + 3418 p T^{7} + 197 p^{2} T^{8} - 251 p^{3} T^{9} + 4 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 20 T + 6 p T^{2} + 1398 T^{3} + 10342 T^{4} + 62234 T^{5} + 331987 T^{6} + 62234 p T^{7} + 10342 p^{2} T^{8} + 1398 p^{3} T^{9} + 6 p^{5} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + T + p T^{2} )^{6} \)
41 \( 1 - 90 T^{2} + 18 T^{3} + 4410 T^{4} - 810 T^{5} - 194177 T^{6} - 810 p T^{7} + 4410 p^{2} T^{8} + 18 p^{3} T^{9} - 90 p^{4} T^{10} + p^{6} T^{12} \)
43 \( ( 1 - 12 T - 6 T^{2} + 547 T^{3} - 6 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )( 1 + 18 T + 198 T^{2} + 1519 T^{3} + 198 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} ) \)
47 \( 1 + 9 T - 6 T^{2} - 531 T^{3} - 2433 T^{4} + 3438 T^{5} + 104623 T^{6} + 3438 p T^{7} - 2433 p^{2} T^{8} - 531 p^{3} T^{9} - 6 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 - 15 T + 225 T^{2} - 1671 T^{3} + 225 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 14 T - 20 T^{2} - 154 T^{3} + 11666 T^{4} + 35126 T^{5} - 499301 T^{6} + 35126 p T^{7} + 11666 p^{2} T^{8} - 154 p^{3} T^{9} - 20 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 8 T - 114 T^{2} - 342 T^{3} + 13762 T^{4} + 13214 T^{5} - 937217 T^{6} + 13214 p T^{7} + 13762 p^{2} T^{8} - 342 p^{3} T^{9} - 114 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - T - 88 T^{2} - 243 T^{3} + 2035 T^{4} + 14290 T^{5} + 72259 T^{6} + 14290 p T^{7} + 2035 p^{2} T^{8} - 243 p^{3} T^{9} - 88 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 7 T + 15 T^{2} - 599 T^{3} + 15 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 - 19 T + 227 T^{2} - 2143 T^{3} + 227 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 - 5 T - 138 T^{2} + 123 T^{3} + 11347 T^{4} + 21118 T^{5} - 1048937 T^{6} + 21118 p T^{7} + 11347 p^{2} T^{8} + 123 p^{3} T^{9} - 138 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 2 T - 182 T^{2} - 2 T^{3} + 18788 T^{4} + 13564 T^{5} - 1721225 T^{6} + 13564 p T^{7} + 18788 p^{2} T^{8} - 2 p^{3} T^{9} - 182 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
89 \( ( 1 - 9 T + 225 T^{2} - 1611 T^{3} + 225 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 + 28 T + 281 T^{2} + 2724 T^{3} + 45178 T^{4} + 388196 T^{5} + 2169217 T^{6} + 388196 p T^{7} + 45178 p^{2} T^{8} + 2724 p^{3} T^{9} + 281 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.65676320103348281024123395156, −4.42835164209663311732385833944, −4.42520291332743537349855257992, −4.29835577989414135198562830530, −3.94787270898438915038595945621, −3.68633915291335199972451912057, −3.62101782985632441679691242859, −3.55269637601442363107660871320, −3.35621068185834193985429926033, −3.14007996690592302023582866215, −3.00248802034643766666569589481, −2.99855161242341093908000291839, −2.64093695393685550899483372868, −2.49167177381983379697760297021, −2.13335514978875195336900468643, −2.08598090263798313458014904526, −1.92711509274754410780112416266, −1.83355682016759914144432080833, −1.76064568658735599323660595464, −1.29067673142139025170098219401, −1.12683147405617377670986172584, −0.847002884573879445483778107988, −0.73214290956457534230336094809, −0.51816314391958849534831881699, −0.13990867595718656587888165101, 0.13990867595718656587888165101, 0.51816314391958849534831881699, 0.73214290956457534230336094809, 0.847002884573879445483778107988, 1.12683147405617377670986172584, 1.29067673142139025170098219401, 1.76064568658735599323660595464, 1.83355682016759914144432080833, 1.92711509274754410780112416266, 2.08598090263798313458014904526, 2.13335514978875195336900468643, 2.49167177381983379697760297021, 2.64093695393685550899483372868, 2.99855161242341093908000291839, 3.00248802034643766666569589481, 3.14007996690592302023582866215, 3.35621068185834193985429926033, 3.55269637601442363107660871320, 3.62101782985632441679691242859, 3.68633915291335199972451912057, 3.94787270898438915038595945621, 4.29835577989414135198562830530, 4.42520291332743537349855257992, 4.42835164209663311732385833944, 4.65676320103348281024123395156

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

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