↑

Properties

Label 24-876e12-1.1-c0e12-0-0
Degree $24$
Conductor $2.042\times 10^{35}$
Sign $1$
Analytic cond. $4.87455\times 10^{-5}$
Root an. cond. $0.661196$
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  + 3·9-s + 6·31-s − 12·61-s − 6·67-s + 3·81-s − 12·103-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 + 229-s + 233-s + ⋯
L(s)  = 1  + 3·9-s + 6·31-s − 12·61-s − 6·67-s + 3·81-s − 12·103-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 + 229-s + 233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 73^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 73^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 3^{12} \cdot 73^{12}\)
Sign: $1$
Analytic conductor: \(4.87455\times 10^{-5}\)
Root analytic conductor: \(0.661196\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 3^{12} \cdot 73^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6068392770\)
\(L(\frac12)\) \(\approx\) \(0.6068392770\)
\(L(1)\) 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 \( ( 1 - T^{2} + T^{4} )^{3} \)
73 \( ( 1 + T^{2} )^{6} \)
good5 \( 1 - T^{12} + T^{24} \)
7 \( ( 1 - T^{3} + T^{6} )^{2}( 1 - T^{6} + T^{12} ) \)
11 \( 1 - T^{12} + T^{24} \)
13 \( ( 1 - T^{3} + T^{6} )^{2}( 1 - T^{6} + T^{12} ) \)
17 \( ( 1 - T^{4} + T^{8} )^{3} \)
19 \( ( 1 - T^{2} + T^{4} )^{3}( 1 - T^{6} + T^{12} ) \)
23 \( ( 1 - T^{6} + T^{12} )^{2} \)
29 \( 1 - T^{12} + T^{24} \)
31 \( ( 1 - T + T^{2} )^{6}( 1 - T^{6} + T^{12} ) \)
37 \( ( 1 - T^{6} + T^{12} )^{2} \)
41 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
43 \( ( 1 + T^{3} + T^{6} )^{2}( 1 - T^{6} + T^{12} ) \)
47 \( 1 - T^{12} + T^{24} \)
53 \( 1 - T^{12} + T^{24} \)
59 \( 1 - T^{12} + T^{24} \)
61 \( ( 1 + T )^{12}( 1 + T^{3} + T^{6} )^{2} \)
67 \( ( 1 + T + T^{2} )^{6}( 1 - T^{3} + T^{6} )^{2} \)
71 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
79 \( ( 1 + T^{2} )^{6}( 1 - T^{6} + T^{12} ) \)
83 \( ( 1 + T^{4} )^{6} \)
89 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
97 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.64212797802653477319939507111, −3.27292637865475448113470782937, −3.22237672484143143401125251078, −3.15089782038192520479375227580, −3.06058257017169321997470358584, −3.02382324078786163815973953215, −2.92696092277547738590267655081, −2.85713821964096419438579830728, −2.81057924664315450848766762383, −2.73592855306068369472590667778, −2.60490779707838394867947567037, −2.58705219574604861753093614846, −2.45438826696294529853349848613, −2.19364054274912134156601082890, −2.10229552951848598092955645971, −1.80949748560620807042071841489, −1.62620754833653079943951388609, −1.52549644750459409713258768050, −1.49141120245201680494076278535, −1.48343607367381098088752702527, −1.47154917226766950370732396638, −1.39011864945304327806498006748, −1.22313483457478200214839647079, −1.03258439941285947408298456503, −0.59422924048075481769536626056, 0.59422924048075481769536626056, 1.03258439941285947408298456503, 1.22313483457478200214839647079, 1.39011864945304327806498006748, 1.47154917226766950370732396638, 1.48343607367381098088752702527, 1.49141120245201680494076278535, 1.52549644750459409713258768050, 1.62620754833653079943951388609, 1.80949748560620807042071841489, 2.10229552951848598092955645971, 2.19364054274912134156601082890, 2.45438826696294529853349848613, 2.58705219574604861753093614846, 2.60490779707838394867947567037, 2.73592855306068369472590667778, 2.81057924664315450848766762383, 2.85713821964096419438579830728, 2.92696092277547738590267655081, 3.02382324078786163815973953215, 3.06058257017169321997470358584, 3.15089782038192520479375227580, 3.22237672484143143401125251078, 3.27292637865475448113470782937, 3.64212797802653477319939507111

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

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