Properties

Label 24-2001e12-1.1-c0e12-0-1
Degree $24$
Conductor $4.121\times 10^{39}$
Sign $1$
Analytic cond. $0.983672$
Root an. cond. $0.999314$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·3-s − 5·4-s − 4·6-s − 12·8-s + 9-s + 10·12-s + 15·16-s + 2·18-s + 24·24-s − 2·25-s + 2·31-s + 42·32-s − 5·36-s − 2·41-s + 2·47-s − 30·48-s − 2·49-s − 4·50-s + 4·62-s − 35·64-s − 12·72-s + 2·73-s + 4·75-s − 4·82-s − 4·93-s + 4·94-s + ⋯
L(s)  = 1  + 2·2-s − 2·3-s − 5·4-s − 4·6-s − 12·8-s + 9-s + 10·12-s + 15·16-s + 2·18-s + 24·24-s − 2·25-s + 2·31-s + 42·32-s − 5·36-s − 2·41-s + 2·47-s − 30·48-s − 2·49-s − 4·50-s + 4·62-s − 35·64-s − 12·72-s + 2·73-s + 4·75-s − 4·82-s − 4·93-s + 4·94-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 23^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 23^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{12} \cdot 23^{12} \cdot 29^{12}\)
Sign: $1$
Analytic conductor: \(0.983672\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{12} \cdot 23^{12} \cdot 29^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1290584732\)
\(L(\frac12)\) \(\approx\) \(0.1290584732\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
23 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
29 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
good2 \( ( 1 + T^{2} )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
5 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
11 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
13 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
17 \( ( 1 + T^{4} )^{6} \)
19 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
37 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
41 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
43 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
53 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
59 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
61 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
67 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
79 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
83 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
89 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
97 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.25909593652186449835958884096, −3.18656055333057495945095168921, −2.97453377920416241530484285525, −2.90724172506648742573410154711, −2.86376255901029625772121901024, −2.77208973828410978020221550142, −2.69325925309115995567696262578, −2.54032805458544618333110414543, −2.33713761154925118500050241129, −2.27338083380008952489515309338, −2.09124909154696315457701405354, −2.07233789720136915399957026876, −2.04307029592682950993425709034, −1.80996401662873055210617973241, −1.70772850594626610419998659872, −1.48041989502667604295017630805, −1.33062390745457513506420992777, −1.27214741778403983367917596258, −1.20406104806879388650302671359, −1.17947667341555740482404024951, −1.16694647723800286705243893062, −0.72271285857536692083285843453, −0.47403144761599708117908831176, −0.46996291595628832652183201979, −0.40056080148540021042572560132, 0.40056080148540021042572560132, 0.46996291595628832652183201979, 0.47403144761599708117908831176, 0.72271285857536692083285843453, 1.16694647723800286705243893062, 1.17947667341555740482404024951, 1.20406104806879388650302671359, 1.27214741778403983367917596258, 1.33062390745457513506420992777, 1.48041989502667604295017630805, 1.70772850594626610419998659872, 1.80996401662873055210617973241, 2.04307029592682950993425709034, 2.07233789720136915399957026876, 2.09124909154696315457701405354, 2.27338083380008952489515309338, 2.33713761154925118500050241129, 2.54032805458544618333110414543, 2.69325925309115995567696262578, 2.77208973828410978020221550142, 2.86376255901029625772121901024, 2.90724172506648742573410154711, 2.97453377920416241530484285525, 3.18656055333057495945095168921, 3.25909593652186449835958884096

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

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