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Properties

Label 12-3800e6-1.1-c0e6-0-6
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  + 6·3-s − 8-s + 21·9-s − 6·24-s + 55·27-s + 6·41-s − 3·49-s − 3·59-s − 3·67-s − 21·72-s − 3·73-s + 120·81-s − 3·97-s − 6·107-s + 36·123-s + 127-s + 131-s + 137-s + 139-s − 18·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 18·177-s + ⋯
L(s)  = 1  + 6·3-s − 8-s + 21·9-s − 6·24-s + 55·27-s + 6·41-s − 3·49-s − 3·59-s − 3·67-s − 21·72-s − 3·73-s + 120·81-s − 3·97-s − 6·107-s + 36·123-s + 127-s + 131-s + 137-s + 139-s − 18·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 18·177-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

\(L(\frac{1}{2})\) \(\approx\) \(36.09970676\)
\(L(\frac12)\) \(\approx\) \(36.09970676\)
\(L(1)\) 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 \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
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 \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
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 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
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

−4.45093724303031218212184007107, −4.30842202658855752992176899649, −4.06458943427970304346551286714, −4.03256619496704889360884073084, −4.02159505318365359416020440057, −3.85735357939733301705984748299, −3.79334936878745012956437526042, −3.32020516708984746301725260119, −3.24985887428549356795438023698, −3.09218346791879366697914252569, −3.06188747286456195022491006060, −3.05293389421121208712818099326, −2.88605860153871773419304475682, −2.68580524566222964331829090725, −2.54126919979046509345881878895, −2.50227271882891344922748469175, −2.40187061466848050513583326748, −2.01294407397334544062040926801, −1.77345951636550670047352189157, −1.74198064724100441102922874142, −1.56323275383286706708476674064, −1.50786051014669123445812855798, −1.36726342816701515364594699614, −0.893618540191142883717432693184, −0.836253712779650796931991223458, 0.836253712779650796931991223458, 0.893618540191142883717432693184, 1.36726342816701515364594699614, 1.50786051014669123445812855798, 1.56323275383286706708476674064, 1.74198064724100441102922874142, 1.77345951636550670047352189157, 2.01294407397334544062040926801, 2.40187061466848050513583326748, 2.50227271882891344922748469175, 2.54126919979046509345881878895, 2.68580524566222964331829090725, 2.88605860153871773419304475682, 3.05293389421121208712818099326, 3.06188747286456195022491006060, 3.09218346791879366697914252569, 3.24985887428549356795438023698, 3.32020516708984746301725260119, 3.79334936878745012956437526042, 3.85735357939733301705984748299, 4.02159505318365359416020440057, 4.03256619496704889360884073084, 4.06458943427970304346551286714, 4.30842202658855752992176899649, 4.45093724303031218212184007107

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

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