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Properties

Label 12-3800e6-1.1-c0e6-0-5
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

\(L(\frac{1}{2})\) \(\approx\) \(0.4373567200\)
\(L(\frac12)\) \(\approx\) \(0.4373567200\)
\(L(1)\) 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

−4.65588623403090266911680122197, −4.53440605318215059845970878353, −4.45452884919507975297115695901, −4.38417401003492059285509340524, −4.30105730120281373904677862784, −4.08611119879870587017480992804, −4.04082048391047313774880645067, −3.61280674962931583743874246866, −3.37659520587155956345329470465, −3.37616031694706102480738190385, −3.32494972547290781847087915164, −3.28108941571175036622815203704, −2.85705350107394372737111342598, −2.26514056137762521908302021417, −2.24588040020576349407323000242, −2.15419658212726388349431128931, −2.05552982932751691187882542096, −1.83259384862959404810141391154, −1.67424212192426977264743411018, −1.42759487613721642894819643118, −1.15853950373466430246170786952, −0.856843530204907589373333334017, −0.75136841781040699102854412938, −0.73053505432368097083646628390, −0.61065719934470144284918960642, 0.61065719934470144284918960642, 0.73053505432368097083646628390, 0.75136841781040699102854412938, 0.856843530204907589373333334017, 1.15853950373466430246170786952, 1.42759487613721642894819643118, 1.67424212192426977264743411018, 1.83259384862959404810141391154, 2.05552982932751691187882542096, 2.15419658212726388349431128931, 2.24588040020576349407323000242, 2.26514056137762521908302021417, 2.85705350107394372737111342598, 3.28108941571175036622815203704, 3.32494972547290781847087915164, 3.37616031694706102480738190385, 3.37659520587155956345329470465, 3.61280674962931583743874246866, 4.04082048391047313774880645067, 4.08611119879870587017480992804, 4.30105730120281373904677862784, 4.38417401003492059285509340524, 4.45452884919507975297115695901, 4.53440605318215059845970878353, 4.65588623403090266911680122197

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

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