↑

Properties

Label 12-3087e6-1.1-c0e6-0-0
Degree $12$
Conductor $8.654\times 10^{20}$
Sign $1$
Analytic cond. $13.3709$
Root an. cond. $1.24121$
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  + 4-s − 25-s − 5·37-s + 2·43-s + 7·61-s − 2·67-s − 2·79-s − 100-s + 2·109-s + 121-s + 127-s + 131-s + 137-s + 139-s − 5·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 2·172-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 4-s − 25-s − 5·37-s + 2·43-s + 7·61-s − 2·67-s − 2·79-s − 100-s + 2·109-s + 121-s + 127-s + 131-s + 137-s + 139-s − 5·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 2·172-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 7^{18}\)
Sign: $1$
Analytic conductor: \(13.3709\)
Root analytic conductor: \(1.24121\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 7^{18} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.620162802\)
\(L(\frac12)\) \(\approx\) \(1.620162802\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−4.79298121973708606324202419459, −4.41100102111797389378070718780, −4.31860051209683860729003095577, −4.24594447658004309516263897647, −4.09900960360674272248422692083, −3.87552202253160805216859327269, −3.86848993276932510568825753612, −3.78372021683869238576534419186, −3.42447369998823907513511783966, −3.38494677718394018710507325191, −3.25329022633987845168302629307, −3.18517836467439054961608806716, −2.77620438531652123667595692004, −2.74741729824264630179591315228, −2.42791969345345221051665932250, −2.30123179402432426061178744675, −2.28225471783727325949791589716, −2.10349709561936490900434023998, −2.07170660714828375913171958975, −1.57521803955966105697309942107, −1.44861073586819228529186587470, −1.30833286851319213265977639857, −1.28961368326469878969934678234, −0.68966068024091917362560834914, −0.42050173523642512106398203375, 0.42050173523642512106398203375, 0.68966068024091917362560834914, 1.28961368326469878969934678234, 1.30833286851319213265977639857, 1.44861073586819228529186587470, 1.57521803955966105697309942107, 2.07170660714828375913171958975, 2.10349709561936490900434023998, 2.28225471783727325949791589716, 2.30123179402432426061178744675, 2.42791969345345221051665932250, 2.74741729824264630179591315228, 2.77620438531652123667595692004, 3.18517836467439054961608806716, 3.25329022633987845168302629307, 3.38494677718394018710507325191, 3.42447369998823907513511783966, 3.78372021683869238576534419186, 3.86848993276932510568825753612, 3.87552202253160805216859327269, 4.09900960360674272248422692083, 4.24594447658004309516263897647, 4.31860051209683860729003095577, 4.41100102111797389378070718780, 4.79298121973708606324202419459

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

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