Properties

Label 12-882e6-1.1-c1e6-0-0
Degree $12$
Conductor $4.708\times 10^{17}$
Sign $1$
Analytic cond. $122032.$
Root an. cond. $2.65382$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s − 4·3-s + 3·4-s + 5-s − 12·6-s − 2·8-s + 6·9-s + 3·10-s − 11-s − 12·12-s + 8·13-s − 4·15-s − 9·16-s + 8·17-s + 18·18-s + 6·19-s + 3·20-s − 3·22-s − 7·23-s + 8·24-s + 9·25-s + 24·26-s − 5·27-s − 5·29-s − 12·30-s + 20·31-s − 9·32-s + ⋯
L(s)  = 1  + 2.12·2-s − 2.30·3-s + 3/2·4-s + 0.447·5-s − 4.89·6-s − 0.707·8-s + 2·9-s + 0.948·10-s − 0.301·11-s − 3.46·12-s + 2.21·13-s − 1.03·15-s − 9/4·16-s + 1.94·17-s + 4.24·18-s + 1.37·19-s + 0.670·20-s − 0.639·22-s − 1.45·23-s + 1.63·24-s + 9/5·25-s + 4.70·26-s − 0.962·27-s − 0.928·29-s − 2.19·30-s + 3.59·31-s − 1.59·32-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{12} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(122032.\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{12} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5343954803\)
\(L(\frac12)\) \(\approx\) \(0.5343954803\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + 10 T^{2} + 7 p T^{3} + 10 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
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

−5.39983885414039468240922200254, −5.21015479924420109141562474898, −5.02841739944039028659275330582, −4.92868627411304202468032820975, −4.85392502338510091964892379810, −4.55834660338365512276120533092, −4.53850132453205447425762157160, −4.50191822827864400308483692247, −4.04831195630895217181137727373, −3.81996255194929187714955641137, −3.76583840480576380852483470540, −3.49564148819439181113393299090, −3.48578198654754047254959318870, −3.07370583650086279752459969745, −3.00900437444854330808116561329, −2.96027653085674315588026649525, −2.88341207738596044931861776223, −2.37914616324586324900511781359, −2.04576080051073008084162770614, −1.64015263602905558585596955921, −1.60752441804350838460588554488, −1.14794469468481374486503876194, −1.13928234191782547831801909414, −0.76691125660843399200042336221, −0.10828789570721755928625206493, 0.10828789570721755928625206493, 0.76691125660843399200042336221, 1.13928234191782547831801909414, 1.14794469468481374486503876194, 1.60752441804350838460588554488, 1.64015263602905558585596955921, 2.04576080051073008084162770614, 2.37914616324586324900511781359, 2.88341207738596044931861776223, 2.96027653085674315588026649525, 3.00900437444854330808116561329, 3.07370583650086279752459969745, 3.48578198654754047254959318870, 3.49564148819439181113393299090, 3.76583840480576380852483470540, 3.81996255194929187714955641137, 4.04831195630895217181137727373, 4.50191822827864400308483692247, 4.53850132453205447425762157160, 4.55834660338365512276120533092, 4.85392502338510091964892379810, 4.92868627411304202468032820975, 5.02841739944039028659275330582, 5.21015479924420109141562474898, 5.39983885414039468240922200254

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

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