↑

Properties

Label 12-259e6-1.1-c0e6-0-0
Degree $12$
Conductor $3.019\times 10^{14}$
Sign $1$
Analytic cond. $4.66381\times 10^{-6}$
Root an. cond. $0.359524$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 3·4-s + 8-s − 6·16-s + 6·32-s + 6·37-s − 6·43-s − 3·53-s + 64-s − 18·74-s − 3·79-s + 18·86-s + 9·106-s − 3·107-s + 6·109-s + 127-s − 9·128-s + 131-s + 137-s + 139-s + 18·148-s + 149-s + 151-s + 157-s + 9·158-s + 163-s + 167-s + ⋯
L(s)  = 1  − 3·2-s + 3·4-s + 8-s − 6·16-s + 6·32-s + 6·37-s − 6·43-s − 3·53-s + 64-s − 18·74-s − 3·79-s + 18·86-s + 9·106-s − 3·107-s + 6·109-s + 127-s − 9·128-s + 131-s + 137-s + 139-s + 18·148-s + 149-s + 151-s + 157-s + 9·158-s + 163-s + 167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(7^{6} \cdot 37^{6}\)
Sign: $1$
Analytic conductor: \(4.66381\times 10^{-6}\)
Root analytic conductor: \(0.359524\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 7^{6} \cdot 37^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02978450161\)
\(L(\frac12)\) \(\approx\) \(0.02978450161\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + T^{3} + T^{6} \)
37 \( ( 1 - T )^{6} \)
good2 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
3 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
5 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
11 \( ( 1 + T^{3} + T^{6} )^{2} \)
13 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
17 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
19 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
23 \( ( 1 + T^{3} + T^{6} )^{2} \)
29 \( ( 1 + T^{3} + T^{6} )^{2} \)
31 \( ( 1 - T )^{6}( 1 + T )^{6} \)
41 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
43 \( ( 1 + T + T^{2} )^{6} \)
47 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
53 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
59 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
61 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
67 \( ( 1 + T^{3} + T^{6} )^{2} \)
71 \( ( 1 + T^{3} + T^{6} )^{2} \)
73 \( ( 1 - T )^{6}( 1 + T )^{6} \)
79 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
83 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
89 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
97 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.15494447985276402678409082636, −6.85487839228986120597030478075, −6.48187337448916771041516772759, −6.32369882085248349778769295928, −6.27860614289940192975846153597, −6.01377372816088255453350971753, −5.94927726194429941332605994943, −5.63446707641871008442846432135, −5.25150215346516643195060143445, −5.06222226129081141559312654228, −4.75444078955722594782466586454, −4.71737765910970130568569540391, −4.61988957063815958670337173867, −4.45438005960773995100990374074, −4.10188567479927887196067622094, −3.73594235241170141075749784911, −3.70733433619797663883442555929, −3.13783923738109022543122143536, −3.04231052666919472943502389079, −2.90648244531407647026125572368, −2.33817136102532053089983885059, −2.00780982444579923850954530742, −1.79620722045902103835867544304, −1.34742853723878219600968941561, −1.04234308746983937350826231641, 1.04234308746983937350826231641, 1.34742853723878219600968941561, 1.79620722045902103835867544304, 2.00780982444579923850954530742, 2.33817136102532053089983885059, 2.90648244531407647026125572368, 3.04231052666919472943502389079, 3.13783923738109022543122143536, 3.70733433619797663883442555929, 3.73594235241170141075749784911, 4.10188567479927887196067622094, 4.45438005960773995100990374074, 4.61988957063815958670337173867, 4.71737765910970130568569540391, 4.75444078955722594782466586454, 5.06222226129081141559312654228, 5.25150215346516643195060143445, 5.63446707641871008442846432135, 5.94927726194429941332605994943, 6.01377372816088255453350971753, 6.27860614289940192975846153597, 6.32369882085248349778769295928, 6.48187337448916771041516772759, 6.85487839228986120597030478075, 7.15494447985276402678409082636

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

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