Properties

Degree $12$
Conductor $3.075\times 10^{13}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 3-s + 8·4-s − 4·6-s + 10·8-s − 2·9-s − 20·11-s − 8·12-s − 8·18-s + 4·19-s − 80·22-s − 18·23-s − 10·24-s + 13·25-s + 3·27-s − 28·32-s + 20·33-s − 16·36-s + 16·38-s − 160·44-s − 72·46-s + 46·47-s − 34·49-s + 52·50-s + 12·54-s − 4·57-s − 10·59-s + ⋯
L(s)  = 1  + 2.82·2-s − 0.577·3-s + 4·4-s − 1.63·6-s + 3.53·8-s − 2/3·9-s − 6.03·11-s − 2.30·12-s − 1.88·18-s + 0.917·19-s − 17.0·22-s − 3.75·23-s − 2.04·24-s + 13/5·25-s + 0.577·27-s − 4.94·32-s + 3.48·33-s − 8/3·36-s + 2.59·38-s − 24.1·44-s − 10.6·46-s + 6.70·47-s − 4.85·49-s + 7.35·50-s + 1.63·54-s − 0.529·57-s − 1.30·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 59^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 59^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 59^{6}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{177} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 59^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.693946\)
\(L(\frac12)\) \(\approx\) \(0.693946\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T + p T^{2} + 2 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
59 \( 1 + 10 T + 85 T^{2} + 732 T^{3} + 85 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
good2 \( ( 1 - p T + p T^{2} - T^{3} + p^{2} T^{4} - p^{3} T^{5} + p^{3} T^{6} )^{2} \)
5 \( 1 - 13 T^{2} + 106 T^{4} - 637 T^{6} + 106 p^{2} T^{8} - 13 p^{4} T^{10} + p^{6} T^{12} \)
7 \( ( 1 + 17 T^{2} + T^{3} + 17 p T^{4} + p^{3} T^{6} )^{2} \)
11 \( ( 1 + 10 T + 60 T^{2} + 234 T^{3} + 60 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 - 20 T^{2} + 516 T^{4} - 6542 T^{6} + 516 p^{2} T^{8} - 20 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 75 T^{2} + 2696 T^{4} - 57772 T^{6} + 2696 p^{2} T^{8} - 75 p^{4} T^{10} + p^{6} T^{12} \)
19 \( ( 1 - 2 T + 31 T^{2} - 113 T^{3} + 31 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( ( 1 + 9 T + 4 p T^{2} + 428 T^{3} + 4 p^{2} T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( 1 - 120 T^{2} + 6965 T^{4} - 249781 T^{6} + 6965 p^{2} T^{8} - 120 p^{4} T^{10} + p^{6} T^{12} \)
31 \( 1 - 71 T^{2} + 3744 T^{4} - 137216 T^{6} + 3744 p^{2} T^{8} - 71 p^{4} T^{10} + p^{6} T^{12} \)
37 \( 1 + T^{2} + 1068 T^{4} + 37816 T^{6} + 1068 p^{2} T^{8} + p^{4} T^{10} + p^{6} T^{12} \)
41 \( 1 - 188 T^{2} + 15985 T^{4} - 816533 T^{6} + 15985 p^{2} T^{8} - 188 p^{4} T^{10} + p^{6} T^{12} \)
43 \( 1 - 52 T^{2} + 5868 T^{4} - 182854 T^{6} + 5868 p^{2} T^{8} - 52 p^{4} T^{10} + p^{6} T^{12} \)
47 \( ( 1 - 23 T + 312 T^{2} - 2568 T^{3} + 312 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( 1 - 197 T^{2} + 19498 T^{4} - 1253909 T^{6} + 19498 p^{2} T^{8} - 197 p^{4} T^{10} + p^{6} T^{12} \)
61 \( 1 - 167 T^{2} + 17204 T^{4} - 1305528 T^{6} + 17204 p^{2} T^{8} - 167 p^{4} T^{10} + p^{6} T^{12} \)
67 \( 1 - 344 T^{2} + 52788 T^{4} - 4582178 T^{6} + 52788 p^{2} T^{8} - 344 p^{4} T^{10} + p^{6} T^{12} \)
71 \( 1 - 120 T^{2} + 132 T^{4} + 542342 T^{6} + 132 p^{2} T^{8} - 120 p^{4} T^{10} + p^{6} T^{12} \)
73 \( 1 - 191 T^{2} + 21072 T^{4} - 1746740 T^{6} + 21072 p^{2} T^{8} - 191 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 + 7 T + 152 T^{2} + 759 T^{3} + 152 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( ( 1 + 25 T + 400 T^{2} + 4164 T^{3} + 400 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( ( 1 - 13 T + 242 T^{2} - 1796 T^{3} + 242 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 240 T^{2} + 15788 T^{4} - 353866 T^{6} + 15788 p^{2} T^{8} - 240 p^{4} T^{10} + p^{6} T^{12} \)
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

−7.15110472524474913082286996705, −7.05584505230003182144693412766, −6.27137909571888206432802433380, −6.14658450996326248975696372143, −6.07714254149424880034628171358, −5.91060337434006145927461493305, −5.73588014074355941509733657907, −5.71879654441117092135157641968, −5.34966373185887616258985106069, −5.05037778064249550871565612209, −4.93177564994334675826330912070, −4.83725319731971049382623901596, −4.79549447942568062278911336079, −4.55846289460673899735296373355, −4.12915229721933757523501965293, −4.02562780776801685586261878273, −3.65602620021831977319715360409, −3.30315047954845361611129105567, −2.88594219572959946677667570769, −2.71795414198315760237398371845, −2.66613238351236441487762617684, −2.62906071688744704892780818425, −2.16529180372202488921351917247, −1.80939368130398762733752978900, −0.25028104011450143750063096256, 0.25028104011450143750063096256, 1.80939368130398762733752978900, 2.16529180372202488921351917247, 2.62906071688744704892780818425, 2.66613238351236441487762617684, 2.71795414198315760237398371845, 2.88594219572959946677667570769, 3.30315047954845361611129105567, 3.65602620021831977319715360409, 4.02562780776801685586261878273, 4.12915229721933757523501965293, 4.55846289460673899735296373355, 4.79549447942568062278911336079, 4.83725319731971049382623901596, 4.93177564994334675826330912070, 5.05037778064249550871565612209, 5.34966373185887616258985106069, 5.71879654441117092135157641968, 5.73588014074355941509733657907, 5.91060337434006145927461493305, 6.07714254149424880034628171358, 6.14658450996326248975696372143, 6.27137909571888206432802433380, 7.05584505230003182144693412766, 7.15110472524474913082286996705

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

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