Properties

Label 24-12e36-1.1-c3e12-0-0
Degree $24$
Conductor $7.088\times 10^{38}$
Sign $1$
Analytic cond. $1.26158\times 10^{24}$
Root an. cond. $10.0972$
Motivic weight $3$
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  + 72·13-s + 558·25-s + 240·37-s + 2.20e3·49-s − 144·61-s + 156·73-s + 516·97-s − 3.21e3·109-s − 6.11e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.24e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 1.53·13-s + 4.46·25-s + 1.06·37-s + 6.41·49-s − 0.302·61-s + 0.250·73-s + 0.540·97-s − 2.82·109-s − 4.59·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 5.65·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{72} \cdot 3^{36}\)
Sign: $1$
Analytic conductor: \(1.26158\times 10^{24}\)
Root analytic conductor: \(10.0972\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{72} \cdot 3^{36} ,\ ( \ : [3/2]^{12} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(4.189986897\)
\(L(\frac12)\) \(\approx\) \(4.189986897\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good5 \( ( 1 - 279 T^{2} + 14742 T^{4} + 9281 p^{3} T^{6} + 14742 p^{6} T^{8} - 279 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
7 \( ( 1 - 1101 T^{2} + 652854 T^{4} - 259162985 T^{6} + 652854 p^{6} T^{8} - 1101 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
11 \( ( 1 + 3057 T^{2} + 6556998 T^{4} + 9226981429 T^{6} + 6556998 p^{6} T^{8} + 3057 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
13 \( ( 1 - 18 T + 3915 T^{2} - 8332 T^{3} + 3915 p^{3} T^{4} - 18 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
17 \( ( 1 - 11946 T^{2} + 72737391 T^{4} - 332667352460 T^{6} + 72737391 p^{6} T^{8} - 11946 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
19 \( ( 1 - 438 p T^{2} + 146058135 T^{4} - 772095813500 T^{6} + 146058135 p^{6} T^{8} - 438 p^{13} T^{10} + p^{18} T^{12} )^{2} \)
23 \( ( 1 + 52314 T^{2} + 1249837599 T^{4} + 18511268917228 T^{6} + 1249837599 p^{6} T^{8} + 52314 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
29 \( ( 1 - 117954 T^{2} + 6353615223 T^{4} - 198355607069564 T^{6} + 6353615223 p^{6} T^{8} - 117954 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
31 \( ( 1 - 45741 T^{2} + 1957054854 T^{4} - 51110094297449 T^{6} + 1957054854 p^{6} T^{8} - 45741 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
37 \( ( 1 - 60 T + 83559 T^{2} + 2089640 T^{3} + 83559 p^{3} T^{4} - 60 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
41 \( ( 1 - 189606 T^{2} + 21529296543 T^{4} - 1820253257865236 T^{6} + 21529296543 p^{6} T^{8} - 189606 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
43 \( ( 1 + 27042 T^{2} + 15896713575 T^{4} + 267144985943548 T^{6} + 15896713575 p^{6} T^{8} + 27042 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
47 \( ( 1 + 377142 T^{2} + 74876518767 T^{4} + 9593174285133748 T^{6} + 74876518767 p^{6} T^{8} + 377142 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
53 \( ( 1 - 398679 T^{2} + 82394238966 T^{4} - 238475483639575 p T^{6} + 82394238966 p^{6} T^{8} - 398679 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
59 \( ( 1 + 275982 T^{2} + 30860919687 T^{4} - 3157264383153500 T^{6} + 30860919687 p^{6} T^{8} + 275982 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
61 \( ( 1 + 36 T + 351135 T^{2} + 84569384 T^{3} + 351135 p^{3} T^{4} + 36 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
67 \( ( 1 - 1341390 T^{2} + 846750752247 T^{4} - 319903491280410596 T^{6} + 846750752247 p^{6} T^{8} - 1341390 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
71 \( ( 1 + 1203654 T^{2} + 624403431231 T^{4} + 229814346307125076 T^{6} + 624403431231 p^{6} T^{8} + 1203654 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
73 \( ( 1 - 39 T + 955110 T^{2} - 15631571 T^{3} + 955110 p^{3} T^{4} - 39 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
79 \( ( 1 - 1716006 T^{2} + 1350264931311 T^{4} - 731914440695115860 T^{6} + 1350264931311 p^{6} T^{8} - 1716006 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
83 \( ( 1 + 1600737 T^{2} + 1256006483670 T^{4} + 724491266389094437 T^{6} + 1256006483670 p^{6} T^{8} + 1600737 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
89 \( ( 1 + 123846 T^{2} + 872233472127 T^{4} - 77031279330160556 T^{6} + 872233472127 p^{6} T^{8} + 123846 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
97 \( ( 1 - 129 T + 1814286 T^{2} - 533624789 T^{3} + 1814286 p^{3} T^{4} - 129 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
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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

−2.51896664057513656690600197160, −2.48114857100512110823836304580, −2.26881522329052005286838142345, −2.19973227312216911459518216595, −2.19467459758766865814649773854, −2.13359299957408491201442718045, −1.95078739028331705739857220041, −1.85323864361747824132731149838, −1.67853455596717881829148258514, −1.54964719658412373330986837559, −1.46954412504691762454721301419, −1.40686949055077822998912849236, −1.37216324887801639381007361635, −1.21543364178056087309040932329, −1.13736796658922203685111261590, −1.01546217870199095180411752574, −0.992774645619208045845945285911, −0.798828659653351034334979721570, −0.72809569125894633211679568591, −0.63539850038735781088529678807, −0.52111469394317473248700534776, −0.50516639902771721452694085964, −0.49622377649177865754757849625, −0.16122126590392843008578906526, −0.04149202180676792522108580899, 0.04149202180676792522108580899, 0.16122126590392843008578906526, 0.49622377649177865754757849625, 0.50516639902771721452694085964, 0.52111469394317473248700534776, 0.63539850038735781088529678807, 0.72809569125894633211679568591, 0.798828659653351034334979721570, 0.992774645619208045845945285911, 1.01546217870199095180411752574, 1.13736796658922203685111261590, 1.21543364178056087309040932329, 1.37216324887801639381007361635, 1.40686949055077822998912849236, 1.46954412504691762454721301419, 1.54964719658412373330986837559, 1.67853455596717881829148258514, 1.85323864361747824132731149838, 1.95078739028331705739857220041, 2.13359299957408491201442718045, 2.19467459758766865814649773854, 2.19973227312216911459518216595, 2.26881522329052005286838142345, 2.48114857100512110823836304580, 2.51896664057513656690600197160

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

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