Properties

Label 12-3e36-1.1-c1e6-0-2
Degree $12$
Conductor $1.501\times 10^{17}$
Sign $1$
Analytic cond. $38907.0$
Root an. cond. $2.41269$
Motivic weight $1$
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  + 21·19-s − 33·37-s + 8·64-s + 21·73-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 + ⋯
L(s)  = 1  + 4.81·19-s − 5.42·37-s + 64-s + 2.45·73-s + 1.14·109-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(3^{36}\)
Sign: $1$
Analytic conductor: \(38907.0\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{729} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{36} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.377878341\)
\(L(\frac12)\) \(\approx\) \(3.377878341\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 - p^{3} T^{6} + p^{6} T^{12} \)
5 \( 1 - p^{3} T^{6} + p^{6} T^{12} \)
7 \( ( 1 - 17 T^{3} + p^{3} T^{6} )( 1 + 37 T^{3} + p^{3} T^{6} ) \)
11 \( 1 - p^{3} T^{6} + p^{6} T^{12} \)
13 \( ( 1 - 89 T^{3} + p^{3} T^{6} )( 1 + 19 T^{3} + p^{3} T^{6} ) \)
17 \( ( 1 - p T^{2} + p^{2} T^{4} )^{3} \)
19 \( ( 1 - 8 T + p T^{2} )^{3}( 1 + T + p T^{2} )^{3} \)
23 \( 1 - p^{3} T^{6} + p^{6} T^{12} \)
29 \( 1 - p^{3} T^{6} + p^{6} T^{12} \)
31 \( ( 1 + 19 T^{3} + p^{3} T^{6} )( 1 + 289 T^{3} + p^{3} T^{6} ) \)
37 \( ( 1 + T + p T^{2} )^{3}( 1 + 10 T + p T^{2} )^{3} \)
41 \( 1 - p^{3} T^{6} + p^{6} T^{12} \)
43 \( ( 1 - 449 T^{3} + p^{3} T^{6} )( 1 - 71 T^{3} + p^{3} T^{6} ) \)
47 \( 1 - p^{3} T^{6} + p^{6} T^{12} \)
53 \( ( 1 + p T^{2} )^{6} \)
59 \( 1 - p^{3} T^{6} + p^{6} T^{12} \)
61 \( ( 1 - 719 T^{3} + p^{3} T^{6} )( 1 + 901 T^{3} + p^{3} T^{6} ) \)
67 \( ( 1 - 1007 T^{3} + p^{3} T^{6} )( 1 + 127 T^{3} + p^{3} T^{6} ) \)
71 \( ( 1 - p T^{2} + p^{2} T^{4} )^{3} \)
73 \( ( 1 - 17 T + p T^{2} )^{3}( 1 + 10 T + p T^{2} )^{3} \)
79 \( ( 1 - 503 T^{3} + p^{3} T^{6} )( 1 + 1387 T^{3} + p^{3} T^{6} ) \)
83 \( 1 - p^{3} T^{6} + p^{6} T^{12} \)
89 \( ( 1 - p T^{2} + p^{2} T^{4} )^{3} \)
97 \( ( 1 - 1853 T^{3} + p^{3} T^{6} )( 1 + 523 T^{3} + p^{3} T^{6} ) \)
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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

−5.39693270257026677572625769455, −5.19740217125182222965565813465, −5.15651556945074164642761987560, −5.11962502753319739703464465246, −5.02359505910417287553351515004, −4.90893828520675537275892166809, −4.75073829087518742391180276142, −4.35666792485979001349701728828, −3.89218635874035959723193267001, −3.89140524968830114346084977688, −3.62090610091713417808477378267, −3.61496372560854211377731395538, −3.57735746289543123429592203540, −3.36275353103461006961630319987, −3.06786296204435527771850437524, −2.72184952383379176943349457570, −2.60294011361708084539319257101, −2.55910998280373179008796778097, −2.23758673685542541862360751672, −1.65923592381617382130211192189, −1.53827353629654300169931148094, −1.50235869957726741863936480270, −1.21879467525710883604211949684, −0.71330027533948425462725648783, −0.37773775011047904446030288029, 0.37773775011047904446030288029, 0.71330027533948425462725648783, 1.21879467525710883604211949684, 1.50235869957726741863936480270, 1.53827353629654300169931148094, 1.65923592381617382130211192189, 2.23758673685542541862360751672, 2.55910998280373179008796778097, 2.60294011361708084539319257101, 2.72184952383379176943349457570, 3.06786296204435527771850437524, 3.36275353103461006961630319987, 3.57735746289543123429592203540, 3.61496372560854211377731395538, 3.62090610091713417808477378267, 3.89140524968830114346084977688, 3.89218635874035959723193267001, 4.35666792485979001349701728828, 4.75073829087518742391180276142, 4.90893828520675537275892166809, 5.02359505910417287553351515004, 5.11962502753319739703464465246, 5.15651556945074164642761987560, 5.19740217125182222965565813465, 5.39693270257026677572625769455

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

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