Properties

Label 12-2592e6-1.1-c0e6-0-0
Degree $12$
Conductor $3.033\times 10^{20}$
Sign $1$
Analytic cond. $4.68546$
Root an. cond. $1.13735$
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  − 3·11-s + 3·41-s + 3·43-s + 6·59-s + 3·67-s − 3·89-s − 3·97-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 + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  − 3·11-s + 3·41-s + 3·43-s + 6·59-s + 3·67-s − 3·89-s − 3·97-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 + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{30} \cdot 3^{24}\)
Sign: $1$
Analytic conductor: \(4.68546\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{30} \cdot 3^{24} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.468211944\)
\(L(\frac12)\) \(\approx\) \(1.468211944\)
\(L(1)\) 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 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
11 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
13 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
17 \( ( 1 - T^{3} + T^{6} )^{2} \)
19 \( ( 1 - T^{3} + T^{6} )^{2} \)
23 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
29 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
31 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
37 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
41 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
43 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
47 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
53 \( ( 1 - T )^{6}( 1 + T )^{6} \)
59 \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
61 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
67 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
71 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
73 \( ( 1 + T^{3} + T^{6} )^{2} \)
79 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
83 \( ( 1 + T^{3} + T^{6} )^{2} \)
89 \( ( 1 + T )^{6}( 1 - T + T^{2} )^{3} \)
97 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
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

−4.92153951003030610557563753914, −4.63668917642408023719454921272, −4.38825405081167792219324991990, −4.32884160958036496957294819878, −4.22144545673409337907756702901, −4.07969017726788857856444149683, −4.05614778269268557633023384366, −3.82295588319089525710229567989, −3.66864028886608599048794510312, −3.50150686272382424135787698450, −3.18359856974288113098183845063, −3.06809177748042540077109202365, −3.03935522669636885035998363831, −2.59891757200664342494343042608, −2.45775454327244143439376554693, −2.45356546337570507460625161434, −2.38970647387330603299013254435, −2.35623851475733170797214501753, −2.25770758300260994367170359302, −1.61734870254295898889064221325, −1.51146481085355359407134417322, −1.39632014969099906300913460330, −0.821224091531145128149934751758, −0.70077204607085848314467797144, −0.65642082288869425926984251620, 0.65642082288869425926984251620, 0.70077204607085848314467797144, 0.821224091531145128149934751758, 1.39632014969099906300913460330, 1.51146481085355359407134417322, 1.61734870254295898889064221325, 2.25770758300260994367170359302, 2.35623851475733170797214501753, 2.38970647387330603299013254435, 2.45356546337570507460625161434, 2.45775454327244143439376554693, 2.59891757200664342494343042608, 3.03935522669636885035998363831, 3.06809177748042540077109202365, 3.18359856974288113098183845063, 3.50150686272382424135787698450, 3.66864028886608599048794510312, 3.82295588319089525710229567989, 4.05614778269268557633023384366, 4.07969017726788857856444149683, 4.22144545673409337907756702901, 4.32884160958036496957294819878, 4.38825405081167792219324991990, 4.63668917642408023719454921272, 4.92153951003030610557563753914

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

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