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Properties

Label 12-54e12-1.1-c0e6-0-11
Degree $12$
Conductor $6.148\times 10^{20}$
Sign $1$
Analytic cond. $9.49878$
Root an. cond. $1.20634$
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  + 3·19-s + 3·37-s + 3·73-s + 12·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯
L(s)  = 1  + 3·19-s + 3·37-s + 3·73-s + 12·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 3^{36}\)
Sign: $1$
Analytic conductor: \(9.49878\)
Root analytic conductor: \(1.20634\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 3^{36} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.653513996\)
\(L(\frac12)\) \(\approx\) \(2.653513996\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
7 \( ( 1 + T^{3} + T^{6} )^{2} \)
11 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
13 \( ( 1 + T^{3} + T^{6} )^{2} \)
17 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
19 \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \)
23 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
29 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
31 \( ( 1 + T^{3} + T^{6} )^{2} \)
37 \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \)
41 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
43 \( ( 1 + T^{3} + T^{6} )^{2} \)
47 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
53 \( ( 1 - T )^{6}( 1 + T )^{6} \)
59 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
61 \( ( 1 + T^{3} + T^{6} )^{2} \)
67 \( ( 1 + T^{3} + T^{6} )^{2} \)
71 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
73 \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \)
79 \( ( 1 + T^{3} + T^{6} )^{2} \)
83 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
89 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
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

−4.85031941356895452327321834122, −4.72985442069881502286500660146, −4.30285893133009566131719603764, −4.25088937548966989785406084900, −4.24286962462135214257260730499, −4.02532594608734947625392261060, −3.99912509535050093962770833822, −3.56119742818984898967014802761, −3.36171007752057881481046274668, −3.35365478073951625119845304904, −3.28151272070552999578360951692, −3.20537021822583158928248789956, −3.16830575505932132823356250452, −2.66962827993594456918450597571, −2.60014746838821733989875094774, −2.25909532123487143564689760442, −2.23657376074035047292492061673, −2.21064625364155192216402743476, −2.11645772816230251937146521598, −1.49173975958224186138071253506, −1.47939413894260555533063065734, −1.26202861153496730637493813292, −0.926455049033312825485381457739, −0.826187442566461644081498312610, −0.68227829160933516685621284413, 0.68227829160933516685621284413, 0.826187442566461644081498312610, 0.926455049033312825485381457739, 1.26202861153496730637493813292, 1.47939413894260555533063065734, 1.49173975958224186138071253506, 2.11645772816230251937146521598, 2.21064625364155192216402743476, 2.23657376074035047292492061673, 2.25909532123487143564689760442, 2.60014746838821733989875094774, 2.66962827993594456918450597571, 3.16830575505932132823356250452, 3.20537021822583158928248789956, 3.28151272070552999578360951692, 3.35365478073951625119845304904, 3.36171007752057881481046274668, 3.56119742818984898967014802761, 3.99912509535050093962770833822, 4.02532594608734947625392261060, 4.24286962462135214257260730499, 4.25088937548966989785406084900, 4.30285893133009566131719603764, 4.72985442069881502286500660146, 4.85031941356895452327321834122

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

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