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Properties

Label 12-148e6-1.1-c0e6-0-0
Degree $12$
Conductor $1.051\times 10^{13}$
Sign $1$
Analytic cond. $1.62372\times 10^{-7}$
Root an. cond. $0.271774$
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  − 3·5-s − 8-s − 3·17-s + 3·25-s + 3·40-s − 3·41-s + 6·61-s − 6·73-s + 9·85-s + 6·89-s − 3·109-s − 3·121-s + 125-s + 127-s + 131-s + 3·136-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 3·5-s − 8-s − 3·17-s + 3·25-s + 3·40-s − 3·41-s + 6·61-s − 6·73-s + 9·85-s + 6·89-s − 3·109-s − 3·121-s + 125-s + 127-s + 131-s + 3·136-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 37^{6}\)
Sign: $1$
Analytic conductor: \(1.62372\times 10^{-7}\)
Root analytic conductor: \(0.271774\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 37^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04410552137\)
\(L(\frac12)\) \(\approx\) \(0.04410552137\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−7.65962023900116921280015867960, −7.22126222370582224323610829624, −7.19586654304967017284530179285, −7.03187465897216676542790962941, −6.86497581257609188338958934217, −6.80232339539420751897729334144, −6.33467671870468831117086546064, −6.29270531474065303935794665906, −6.08221636703777934330082755136, −5.74595973402176295464707050931, −5.50316130092364474241279526376, −5.14795358699906242570917458909, −4.99794529017152063699270491463, −4.81760476795408907133508029574, −4.49932666237593842736732035913, −4.29285617711419808267934528886, −4.05554010026699310381168794944, −3.76795293971872327412748300551, −3.73765694704081880445300521503, −3.48199219988604857120472306775, −3.21211670025834868401403851132, −2.60771914128348356188111894493, −2.58179836465354088901631358036, −2.13734296270140680933252114403, −1.63750977340220139756058364922, 1.63750977340220139756058364922, 2.13734296270140680933252114403, 2.58179836465354088901631358036, 2.60771914128348356188111894493, 3.21211670025834868401403851132, 3.48199219988604857120472306775, 3.73765694704081880445300521503, 3.76795293971872327412748300551, 4.05554010026699310381168794944, 4.29285617711419808267934528886, 4.49932666237593842736732035913, 4.81760476795408907133508029574, 4.99794529017152063699270491463, 5.14795358699906242570917458909, 5.50316130092364474241279526376, 5.74595973402176295464707050931, 6.08221636703777934330082755136, 6.29270531474065303935794665906, 6.33467671870468831117086546064, 6.80232339539420751897729334144, 6.86497581257609188338958934217, 7.03187465897216676542790962941, 7.19586654304967017284530179285, 7.22126222370582224323610829624, 7.65962023900116921280015867960

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

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