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Properties

Label 24-3800e12-1.1-c0e12-0-0
Degree $24$
Conductor $9.066\times 10^{42}$
Sign $1$
Analytic cond. $2164.15$
Root an. cond. $1.37711$
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  − 6·41-s + 64-s − 12·89-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 + 239-s + 241-s + 251-s + ⋯
L(s)  = 1  − 6·41-s + 64-s − 12·89-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 + 239-s + 241-s + 251-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 5^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 5^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{36} \cdot 5^{24} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(2164.15\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{36} \cdot 5^{24} \cdot 19^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.413668221\times10^{-5}\)
\(L(\frac12)\) \(\approx\) \(1.413668221\times10^{-5}\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{6} + T^{12} \)
5 \( 1 \)
19 \( ( 1 - T^{3} + T^{6} )^{2} \)
good3 \( ( 1 - T^{6} + T^{12} )^{2} \)
7 \( ( 1 - T^{6} + T^{12} )^{2} \)
11 \( ( 1 + T^{3} + T^{6} )^{4} \)
13 \( ( 1 - T^{6} + T^{12} )^{2} \)
17 \( ( 1 - T^{6} + T^{12} )^{2} \)
23 \( ( 1 - T^{2} + T^{4} )^{3}( 1 - T^{6} + T^{12} ) \)
29 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
31 \( ( 1 - T + T^{2} )^{6}( 1 + T + T^{2} )^{6} \)
37 \( ( 1 - T^{6} + T^{12} )^{2} \)
41 \( ( 1 + T + T^{2} )^{6}( 1 + T^{3} + T^{6} )^{2} \)
43 \( ( 1 - T^{6} + T^{12} )^{2} \)
47 \( ( 1 - T^{6} + T^{12} )^{2} \)
53 \( ( 1 + T^{2} )^{6}( 1 - T^{6} + T^{12} ) \)
59 \( ( 1 - T^{3} + T^{6} )^{4} \)
61 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
67 \( ( 1 - T^{6} + T^{12} )^{2} \)
71 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
73 \( ( 1 - T^{6} + T^{12} )^{2} \)
79 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
83 \( ( 1 - T^{2} + T^{4} )^{6} \)
89 \( ( 1 + T )^{12}( 1 - T^{3} + T^{6} )^{2} \)
97 \( ( 1 - T^{6} + T^{12} )^{2} \)
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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

−2.80366746769176514471843634747, −2.72987191075976230328357279156, −2.57739145307573726846400788545, −2.54552651539510379931434189431, −2.52012124029436483836010643187, −2.50717571169214915264546531428, −2.44514766464527012564157558600, −2.14909085073748212892311383992, −2.04905647114621949797222304320, −1.88485608154986027429513077359, −1.80345465476923585013076675921, −1.71637813427509189513980870520, −1.70870756343875180248521877538, −1.65871990825826427361377317461, −1.60945670089102455026612486123, −1.54056080972528817997209847102, −1.52863128661068222437468926906, −1.44429319361481778165190947211, −1.06351610639851759491848549362, −1.02978445708095806744248117483, −1.02515826655293376180756107969, −0.74751150436094506008877153765, −0.71757907700913506663827882447, −0.40174486317415070186052478663, −0.000832598882436861477407193692, 0.000832598882436861477407193692, 0.40174486317415070186052478663, 0.71757907700913506663827882447, 0.74751150436094506008877153765, 1.02515826655293376180756107969, 1.02978445708095806744248117483, 1.06351610639851759491848549362, 1.44429319361481778165190947211, 1.52863128661068222437468926906, 1.54056080972528817997209847102, 1.60945670089102455026612486123, 1.65871990825826427361377317461, 1.70870756343875180248521877538, 1.71637813427509189513980870520, 1.80345465476923585013076675921, 1.88485608154986027429513077359, 2.04905647114621949797222304320, 2.14909085073748212892311383992, 2.44514766464527012564157558600, 2.50717571169214915264546531428, 2.52012124029436483836010643187, 2.54552651539510379931434189431, 2.57739145307573726846400788545, 2.72987191075976230328357279156, 2.80366746769176514471843634747

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

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