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Properties

Label 12-1083e6-1.1-c0e6-0-1
Degree $12$
Conductor $1.614\times 10^{18}$
Sign $1$
Analytic cond. $0.0249294$
Root an. cond. $0.735178$
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  + 3·7-s + 27-s − 3·31-s + 6·37-s + 6·49-s − 64-s − 3·103-s − 3·121-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 + 3·189-s + 191-s + 193-s + 197-s + 199-s + 211-s − 9·217-s + ⋯
L(s)  = 1  + 3·7-s + 27-s − 3·31-s + 6·37-s + 6·49-s − 64-s − 3·103-s − 3·121-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 + 3·189-s + 191-s + 193-s + 197-s + 199-s + 211-s − 9·217-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(0.0249294\)
Root analytic conductor: \(0.735178\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 19^{12} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.517773055\)
\(L(\frac12)\) \(\approx\) \(1.517773055\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T^{3} + T^{6} \)
19 \( 1 \)
good2 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
5 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
7 \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \)
11 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
13 \( ( 1 - T^{3} + T^{6} )^{2} \)
17 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
23 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
29 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
31 \( ( 1 + T )^{6}( 1 - T + T^{2} )^{3} \)
37 \( ( 1 - T + T^{2} )^{6} \)
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 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
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 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
73 \( ( 1 + T^{3} + T^{6} )^{2} \)
79 \( ( 1 - T^{3} + T^{6} )^{2} \)
83 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
89 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
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

−5.57692685610929721926726518051, −5.17753105298936513836763036697, −5.06428052022004794581004504096, −4.79302019000537351592545218233, −4.78508856821705893406390277564, −4.77985101302363561505232789898, −4.69518512219818483484309990436, −4.18385790514711976676611572899, −4.16451976066171515268737181584, −4.03188928745327089143250677070, −3.92560358245955759693853870326, −3.78155464067630920508368432177, −3.49530902580612170910337908476, −3.36424645775883560310818099890, −2.89130533666237733211322018026, −2.71411987734987024483221287660, −2.53180443418081366840494250159, −2.52112437754072566179686699672, −2.41972740093307188660079537348, −2.06098353047006197875299672972, −1.79830190613010191564818146658, −1.49708702052486119361280431217, −1.26523190841261966974220011341, −1.12564880862962502568071481758, −1.01741565754811547362573533657, 1.01741565754811547362573533657, 1.12564880862962502568071481758, 1.26523190841261966974220011341, 1.49708702052486119361280431217, 1.79830190613010191564818146658, 2.06098353047006197875299672972, 2.41972740093307188660079537348, 2.52112437754072566179686699672, 2.53180443418081366840494250159, 2.71411987734987024483221287660, 2.89130533666237733211322018026, 3.36424645775883560310818099890, 3.49530902580612170910337908476, 3.78155464067630920508368432177, 3.92560358245955759693853870326, 4.03188928745327089143250677070, 4.16451976066171515268737181584, 4.18385790514711976676611572899, 4.69518512219818483484309990436, 4.77985101302363561505232789898, 4.78508856821705893406390277564, 4.79302019000537351592545218233, 5.06428052022004794581004504096, 5.17753105298936513836763036697, 5.57692685610929721926726518051

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

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