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Properties

Label 12-3800e6-1.1-c0e6-0-1
Degree $12$
Conductor $3.011\times 10^{21}$
Sign $1$
Analytic cond. $46.5204$
Root an. cond. $1.37711$
Motivic weight $0$
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  + 3·3-s + 8-s + 3·9-s + 3·24-s − 27-s − 3·41-s − 3·49-s − 3·59-s + 3·67-s + 3·72-s − 6·73-s − 6·81-s + 3·97-s − 3·107-s − 9·123-s + 127-s + 131-s + 137-s + 139-s − 9·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 9·177-s + ⋯
L(s)  = 1  + 3·3-s + 8-s + 3·9-s + 3·24-s − 27-s − 3·41-s − 3·49-s − 3·59-s + 3·67-s + 3·72-s − 6·73-s − 6·81-s + 3·97-s − 3·107-s − 9·123-s + 127-s + 131-s + 137-s + 139-s − 9·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 9·177-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 5^{12} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(46.5204\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 5^{12} \cdot 19^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8255517503\)
\(L(\frac12)\) \(\approx\) \(0.8255517503\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{3} + T^{6} \)
5 \( 1 \)
19 \( 1 + T^{3} + T^{6} \)
good3 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
7 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
11 \( ( 1 + T^{3} + T^{6} )^{2} \)
13 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
17 \( ( 1 - T^{3} + T^{6} )^{2} \)
23 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
29 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
31 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
37 \( ( 1 - T )^{6}( 1 + T )^{6} \)
41 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
43 \( ( 1 - T^{3} + T^{6} )^{2} \)
47 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
53 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
59 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
61 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
67 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
71 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
73 \( ( 1 + T )^{6}( 1 - T^{3} + T^{6} ) \)
79 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
83 \( ( 1 - T^{3} + T^{6} )^{2} \)
89 \( ( 1 + T^{3} + T^{6} )^{2} \)
97 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
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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

−4.52334889933839095118769147383, −4.44115732235627118357036754470, −4.16672468280027395878170425172, −4.15490901312524603297962828187, −4.14867046600230301750422647157, −3.71442409939332230991417682180, −3.42024467240444856171877485873, −3.40855795502559021738878300642, −3.36067603196106313675791407586, −3.27763253592235253427778556120, −3.14879036712836642016871818604, −3.07546935319514519677801288150, −3.01600727090980773279987997249, −2.55907002462088296955854447643, −2.50540148636659873801102332292, −2.35681088816189261826820645353, −2.25963335511131837605370604285, −1.97564559282647654538059889524, −1.86393399172571876064177440809, −1.69981268090546034629691310302, −1.51538650696186285296774844298, −1.45223696666061835205671100235, −1.16414841980705346392993377780, −0.923238296643777423629087500326, −0.14961762086333696360935269126, 0.14961762086333696360935269126, 0.923238296643777423629087500326, 1.16414841980705346392993377780, 1.45223696666061835205671100235, 1.51538650696186285296774844298, 1.69981268090546034629691310302, 1.86393399172571876064177440809, 1.97564559282647654538059889524, 2.25963335511131837605370604285, 2.35681088816189261826820645353, 2.50540148636659873801102332292, 2.55907002462088296955854447643, 3.01600727090980773279987997249, 3.07546935319514519677801288150, 3.14879036712836642016871818604, 3.27763253592235253427778556120, 3.36067603196106313675791407586, 3.40855795502559021738878300642, 3.42024467240444856171877485873, 3.71442409939332230991417682180, 4.14867046600230301750422647157, 4.15490901312524603297962828187, 4.16672468280027395878170425172, 4.44115732235627118357036754470, 4.52334889933839095118769147383

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

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