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Properties

Label 12-2009e6-1.1-c0e6-0-1
Degree $12$
Conductor $6.575\times 10^{19}$
Sign $1$
Analytic cond. $1.01583$
Root an. cond. $1.00130$
Motivic weight $0$
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  + 2-s + 3-s + 4-s + 6-s + 9-s + 12-s − 2·13-s + 17-s + 18-s + 19-s + 23-s − 3·25-s − 2·26-s + 34-s + 36-s + 37-s + 38-s − 2·39-s + 6·41-s − 2·43-s + 46-s + 47-s − 3·50-s + 51-s − 2·52-s + 57-s + 68-s + ⋯
L(s)  = 1  + 2-s + 3-s + 4-s + 6-s + 9-s + 12-s − 2·13-s + 17-s + 18-s + 19-s + 23-s − 3·25-s − 2·26-s + 34-s + 36-s + 37-s + 38-s − 2·39-s + 6·41-s − 2·43-s + 46-s + 47-s − 3·50-s + 51-s − 2·52-s + 57-s + 68-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 41^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 41^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(7^{12} \cdot 41^{6}\)
Sign: $1$
Analytic conductor: \(1.01583\)
Root analytic conductor: \(1.00130\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 7^{12} \cdot 41^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.581340212\)
\(L(\frac12)\) \(\approx\) \(3.581340212\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( ( 1 - T )^{6} \)
good2 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
3 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
5 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
11 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
13 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
17 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
19 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
23 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
29 \( ( 1 - T )^{6}( 1 + T )^{6} \)
31 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
37 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
43 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
47 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
53 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
59 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
61 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
67 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
71 \( ( 1 - T )^{6}( 1 + T )^{6} \)
73 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
79 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
83 \( ( 1 - T )^{6}( 1 + T )^{6} \)
89 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
97 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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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

−5.06076217871056130586479215509, −4.83348513755775113107090671625, −4.62354146552116536650053120354, −4.56913344856577448702948722213, −4.31982938468007056534677439077, −4.20671559624082922823322874354, −3.89404367349217527185079854756, −3.88736094479582720847191612811, −3.73134309398697750689058117393, −3.70260687927945605092683058193, −3.69485738022074715154526430086, −3.13127814975886107732512491956, −3.01503733227752910567772794786, −2.91568693533977308897775536939, −2.71033009321280303841332942819, −2.69947967683693980519869075792, −2.46986856091967004084412632515, −2.40715702296395911562008266405, −2.07155859324925842810246860281, −1.89413076196815578054865504404, −1.89315450443600793012164880268, −1.22366869102097182474999724637, −1.19629192183085145567109304124, −1.18332793290029512309682591489, −0.55953660300118954990355198473, 0.55953660300118954990355198473, 1.18332793290029512309682591489, 1.19629192183085145567109304124, 1.22366869102097182474999724637, 1.89315450443600793012164880268, 1.89413076196815578054865504404, 2.07155859324925842810246860281, 2.40715702296395911562008266405, 2.46986856091967004084412632515, 2.69947967683693980519869075792, 2.71033009321280303841332942819, 2.91568693533977308897775536939, 3.01503733227752910567772794786, 3.13127814975886107732512491956, 3.69485738022074715154526430086, 3.70260687927945605092683058193, 3.73134309398697750689058117393, 3.88736094479582720847191612811, 3.89404367349217527185079854756, 4.20671559624082922823322874354, 4.31982938468007056534677439077, 4.56913344856577448702948722213, 4.62354146552116536650053120354, 4.83348513755775113107090671625, 5.06076217871056130586479215509

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

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