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Properties

Label 24-3648e12-1.1-c0e12-0-1
Degree $24$
Conductor $5.555\times 10^{42}$
Sign $1$
Analytic cond. $1325.99$
Root an. cond. $1.34929$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·13-s − 12·61-s − 6·73-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 15·169-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 + 239-s + ⋯
L(s)  = 1  − 6·13-s − 12·61-s − 6·73-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 15·169-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 + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{72} \cdot 3^{12} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(1325.99\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{72} \cdot 3^{12} \cdot 19^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06895804467\)
\(L(\frac12)\) \(\approx\) \(0.06895804467\)
\(L(1)\) 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 \( 1 - T^{6} + T^{12} \)
19 \( ( 1 - T^{2} + T^{4} )^{3} \)
good5 \( ( 1 - T^{6} + T^{12} )^{2} \)
7 \( ( 1 - T^{6} + T^{12} )^{2} \)
11 \( ( 1 - T^{2} + T^{4} )^{6} \)
13 \( ( 1 + T + T^{2} )^{6}( 1 - T^{3} + T^{6} )^{2} \)
17 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
23 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
29 \( ( 1 - T^{6} + T^{12} )^{2} \)
31 \( ( 1 - T^{6} + T^{12} )^{2} \)
37 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
41 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
43 \( ( 1 - T^{2} + T^{4} )^{3}( 1 - T^{6} + T^{12} ) \)
47 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
53 \( ( 1 - T^{6} + T^{12} )^{2} \)
59 \( ( 1 - T^{6} + T^{12} )^{2} \)
61 \( ( 1 + T )^{12}( 1 + T^{3} + T^{6} )^{2} \)
67 \( ( 1 - T^{2} + T^{4} )^{3}( 1 - T^{6} + T^{12} ) \)
71 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
73 \( ( 1 + T + T^{2} )^{6}( 1 + T^{3} + T^{6} )^{2} \)
79 \( ( 1 + T^{2} )^{6}( 1 - T^{6} + T^{12} ) \)
83 \( ( 1 - T^{2} + T^{4} )^{6} \)
89 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
97 \( ( 1 - T^{3} + T^{6} )^{4} \)
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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

−2.84597320195661121876930233416, −2.73172226013135454565364998374, −2.65170638380728273304523296547, −2.54894108087516940840722986941, −2.48476716922840162176747658170, −2.42109534735181141155613563655, −2.28821040178392228131470459947, −2.16212088572775547732727322272, −2.16196138200293390593186143884, −2.06318955544343781195272499287, −1.88959673372259977127883812035, −1.81156226491768515199137228463, −1.78044517914422612032922062368, −1.77643571190007507971010581234, −1.77494553732643971413754282524, −1.43881524255979826642310482724, −1.31277286202431054847422472556, −1.27211971753040389936757028836, −1.25476101747368007263759586610, −1.17952490456715573258467876838, −1.01084029191155015134064862048, −0.71299851389536603876828768568, −0.49909438124227365581336275656, −0.26632512070614270974989484201, −0.12110128644508897394946465074, 0.12110128644508897394946465074, 0.26632512070614270974989484201, 0.49909438124227365581336275656, 0.71299851389536603876828768568, 1.01084029191155015134064862048, 1.17952490456715573258467876838, 1.25476101747368007263759586610, 1.27211971753040389936757028836, 1.31277286202431054847422472556, 1.43881524255979826642310482724, 1.77494553732643971413754282524, 1.77643571190007507971010581234, 1.78044517914422612032922062368, 1.81156226491768515199137228463, 1.88959673372259977127883812035, 2.06318955544343781195272499287, 2.16196138200293390593186143884, 2.16212088572775547732727322272, 2.28821040178392228131470459947, 2.42109534735181141155613563655, 2.48476716922840162176747658170, 2.54894108087516940840722986941, 2.65170638380728273304523296547, 2.73172226013135454565364998374, 2.84597320195661121876930233416

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

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