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Properties

Label 24-2900e12-1.1-c0e12-0-7
Degree $24$
Conductor $3.538\times 10^{41}$
Sign $1$
Analytic cond. $84.4620$
Root an. cond. $1.20303$
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  + 4-s + 2·9-s − 2·29-s + 2·36-s − 2·49-s + 81-s + 10·109-s − 2·116-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 2·196-s + 197-s + 199-s + ⋯
L(s)  = 1  + 4-s + 2·9-s − 2·29-s + 2·36-s − 2·49-s + 81-s + 10·109-s − 2·116-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 2·196-s + 197-s + 199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{24} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{24} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 5^{24} \cdot 29^{12}\)
Sign: $1$
Analytic conductor: \(84.4620\)
Root analytic conductor: \(1.20303\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 5^{24} \cdot 29^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.337112588\)
\(L(\frac12)\) \(\approx\) \(4.337112588\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
5 \( 1 \)
29 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
good3 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
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 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
13 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
17 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
19 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
31 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
37 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
43 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
47 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
53 \( ( 1 + T^{2} )^{6}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
59 \( ( 1 - T )^{12}( 1 + T )^{12} \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
73 \( ( 1 + T^{2} )^{6}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
79 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
89 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
97 \( ( 1 + T^{2} )^{6}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
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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

−2.89354477155847938049393097569, −2.88022872098787071803419609059, −2.72082238459584502678220577490, −2.66150840717772549584545640728, −2.50001127832388880807566672472, −2.44589667490863549265351284855, −2.24540446869476858915137770271, −2.21319474888230833302046576580, −2.10486036172280048313629193832, −2.09607152163188766682012834071, −2.06367047792344848141167134951, −1.99614485921579952282823801947, −1.93465443486336698042342103250, −1.72564528490600214165943788964, −1.69170884453784750678704747111, −1.56139082742200618817276716191, −1.49482556439290452285954280992, −1.31418100724068826936124697599, −1.23859563387551097199707302950, −1.22225918310802237869203920592, −1.12052348740793572037709647904, −0.78555268750047837993410416102, −0.64635204945446399094113467910, −0.59283236276685902561678745348, −0.48956586045514781450864132719, 0.48956586045514781450864132719, 0.59283236276685902561678745348, 0.64635204945446399094113467910, 0.78555268750047837993410416102, 1.12052348740793572037709647904, 1.22225918310802237869203920592, 1.23859563387551097199707302950, 1.31418100724068826936124697599, 1.49482556439290452285954280992, 1.56139082742200618817276716191, 1.69170884453784750678704747111, 1.72564528490600214165943788964, 1.93465443486336698042342103250, 1.99614485921579952282823801947, 2.06367047792344848141167134951, 2.09607152163188766682012834071, 2.10486036172280048313629193832, 2.21319474888230833302046576580, 2.24540446869476858915137770271, 2.44589667490863549265351284855, 2.50001127832388880807566672472, 2.66150840717772549584545640728, 2.72082238459584502678220577490, 2.88022872098787071803419609059, 2.89354477155847938049393097569

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

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