Properties

Degree 12
Conductor $ 2^{30} \cdot 3^{6} \cdot 5^{12} $
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  − 6·3-s + 21·9-s − 56·27-s + 12·31-s + 8·37-s − 20·41-s + 8·43-s + 6·49-s + 12·53-s − 24·67-s + 8·71-s + 36·79-s + 126·81-s − 32·83-s + 28·89-s − 72·93-s − 40·107-s − 48·111-s + 2·121-s + 120·123-s + 127-s − 48·129-s + 131-s + 137-s + 139-s − 36·147-s + 149-s + ⋯
L(s)  = 1  − 3.46·3-s + 7·9-s − 10.7·27-s + 2.15·31-s + 1.31·37-s − 3.12·41-s + 1.21·43-s + 6/7·49-s + 1.64·53-s − 2.93·67-s + 0.949·71-s + 4.05·79-s + 14·81-s − 3.51·83-s + 2.96·89-s − 7.46·93-s − 3.86·107-s − 4.55·111-s + 2/11·121-s + 10.8·123-s + 0.0887·127-s − 4.22·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.96·147-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{6} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{6} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(12\)
\( N \)  =  \(2^{30} \cdot 3^{6} \cdot 5^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{2400} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((12,\ 2^{30} \cdot 3^{6} \cdot 5^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(1.331243339\)
\(L(\frac12)\)  \(\approx\)  \(1.331243339\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 12. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 11.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + T )^{6} \)
5 \( 1 \)
good7 \( 1 - 6 T^{2} + 47 T^{4} - 500 T^{6} + 47 p^{2} T^{8} - 6 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 2 T^{2} + 87 T^{4} + 4 T^{6} + 87 p^{2} T^{8} - 2 p^{4} T^{10} + p^{6} T^{12} \)
13 \( ( 1 + 11 T^{2} + 16 T^{3} + 11 p T^{4} + p^{3} T^{6} )^{2} \)
17 \( 1 - 2 p T^{2} + 351 T^{4} - 1084 T^{6} + 351 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \)
19 \( 1 - 74 T^{2} + 2647 T^{4} - 60620 T^{6} + 2647 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12} \)
23 \( 1 - 98 T^{2} + 4527 T^{4} - 128636 T^{6} + 4527 p^{2} T^{8} - 98 p^{4} T^{10} + p^{6} T^{12} \)
29 \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{3} \)
31 \( ( 1 - 6 T + 77 T^{2} - 308 T^{3} + 77 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( ( 1 - 4 T + 51 T^{2} - 40 T^{3} + 51 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( ( 1 + 10 T + 87 T^{2} + 588 T^{3} + 87 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 - 4 T + 65 T^{2} - 216 T^{3} + 65 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 82 T^{2} + 7967 T^{4} - 348252 T^{6} + 7967 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \)
53 \( ( 1 - 2 T + p T^{2} )^{6} \)
59 \( 1 - 274 T^{2} + 33911 T^{4} - 2503644 T^{6} + 33911 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \)
61 \( 1 - 110 T^{2} + 10759 T^{4} - 685796 T^{6} + 10759 p^{2} T^{8} - 110 p^{4} T^{10} + p^{6} T^{12} \)
67 \( ( 1 + 4 T + p T^{2} )^{6} \)
71 \( ( 1 - 4 T + 101 T^{2} - 632 T^{3} + 101 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 - 16 T + p T^{2} )^{3}( 1 + 16 T + p T^{2} )^{3} \)
79 \( ( 1 - 18 T + 317 T^{2} - 2908 T^{3} + 317 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( ( 1 + 16 T + 265 T^{2} + 2400 T^{3} + 265 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( ( 1 - 14 T + 263 T^{2} - 2308 T^{3} + 263 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 250 T^{2} + 24143 T^{4} - 1697004 T^{6} + 24143 p^{2} T^{8} - 250 p^{4} T^{10} + p^{6} T^{12} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−4.64240621082944485346952258537, −4.64050388347581146855068942298, −4.43271736156641206076866062438, −4.31689890667564807057564405897, −4.21242215406631015143848368256, −4.09309291493534630291159479639, −3.73551998677796310133921823607, −3.72293630333557068677002239686, −3.52316595033406006892492315126, −3.39323298630402956298264283835, −3.14071082654360387145378940391, −2.96908126547384873592848117038, −2.76867761576719492582758553545, −2.40317585666911572209884099399, −2.38123385107897960401535928205, −2.33764138665036270142862179197, −2.03294365121442277862368646671, −1.66917313509841299267028713508, −1.53049195324949182375990917473, −1.24259243143419065276927415183, −1.21391242591165334624701638485, −1.01278378954041542066262668384, −0.60533226567943684770470271457, −0.37447667192830236477776389283, −0.36165220567442815438782277080, 0.36165220567442815438782277080, 0.37447667192830236477776389283, 0.60533226567943684770470271457, 1.01278378954041542066262668384, 1.21391242591165334624701638485, 1.24259243143419065276927415183, 1.53049195324949182375990917473, 1.66917313509841299267028713508, 2.03294365121442277862368646671, 2.33764138665036270142862179197, 2.38123385107897960401535928205, 2.40317585666911572209884099399, 2.76867761576719492582758553545, 2.96908126547384873592848117038, 3.14071082654360387145378940391, 3.39323298630402956298264283835, 3.52316595033406006892492315126, 3.72293630333557068677002239686, 3.73551998677796310133921823607, 4.09309291493534630291159479639, 4.21242215406631015143848368256, 4.31689890667564807057564405897, 4.43271736156641206076866062438, 4.64050388347581146855068942298, 4.64240621082944485346952258537

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

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