↑

Properties

Label 24-1900e12-1.1-c1e12-0-1
Degree $24$
Conductor $2.213\times 10^{39}$
Sign $1$
Analytic cond. $1.48719\times 10^{14}$
Root an. cond. $3.89507$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s + 8·11-s + 4·17-s − 4·23-s + 28·43-s − 20·47-s + 8·49-s + 16·61-s + 76·73-s − 32·77-s + 34·81-s − 84·83-s − 80·101-s − 16·119-s − 76·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 16·161-s + 163-s + 167-s + 173-s + 179-s + ⋯
L(s)  = 1  − 1.51·7-s + 2.41·11-s + 0.970·17-s − 0.834·23-s + 4.26·43-s − 2.91·47-s + 8/7·49-s + 2.04·61-s + 8.89·73-s − 3.64·77-s + 34/9·81-s − 9.22·83-s − 7.96·101-s − 1.46·119-s − 6.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 1.26·161-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

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

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 5^{24} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(1.48719\times 10^{14}\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 5^{24} \cdot 19^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(9.390438784\)
\(L(\frac12)\) \(\approx\) \(9.390438784\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 - 6 T^{2} + 71 T^{4} - 10484 T^{6} + 71 p^{2} T^{8} - 6 p^{4} T^{10} + p^{6} T^{12} \)
good3 \( 1 - 34 T^{4} + 181 p T^{8} - 5708 T^{12} + 181 p^{5} T^{16} - 34 p^{8} T^{20} + p^{12} T^{24} \)
7 \( ( 1 + 2 T + 2 T^{2} + 6 T^{3} + 15 T^{4} + 52 T^{5} + 92 T^{6} + 52 p T^{7} + 15 p^{2} T^{8} + 6 p^{3} T^{9} + 2 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
11 \( ( 1 - 2 T + 29 T^{2} - 40 T^{3} + 29 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
13 \( 1 - 82 T^{4} - 3425 T^{8} + 7504532 T^{12} - 3425 p^{4} T^{16} - 82 p^{8} T^{20} + p^{12} T^{24} \)
17 \( ( 1 - 2 T + 2 T^{2} - 50 T^{3} - 33 T^{4} + 1172 T^{5} - 1028 T^{6} + 1172 p T^{7} - 33 p^{2} T^{8} - 50 p^{3} T^{9} + 2 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
23 \( ( 1 + 2 T + 2 T^{2} + 86 T^{3} - 273 T^{4} - 3164 T^{5} - 2084 T^{6} - 3164 p T^{7} - 273 p^{2} T^{8} + 86 p^{3} T^{9} + 2 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
29 \( ( 1 + 134 T^{2} + 8471 T^{4} + 312756 T^{6} + 8471 p^{2} T^{8} + 134 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
31 \( ( 1 - 90 T^{2} + 4847 T^{4} - 185164 T^{6} + 4847 p^{2} T^{8} - 90 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
37 \( 1 + 926 T^{4} - 792545 T^{8} - 563012236 T^{12} - 792545 p^{4} T^{16} + 926 p^{8} T^{20} + p^{12} T^{24} \)
41 \( ( 1 - 102 T^{2} + 4031 T^{4} - 117652 T^{6} + 4031 p^{2} T^{8} - 102 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
43 \( ( 1 - 14 T + 98 T^{2} - 802 T^{3} + 5895 T^{4} - 708 p T^{5} + 92 p^{2} T^{6} - 708 p^{2} T^{7} + 5895 p^{2} T^{8} - 802 p^{3} T^{9} + 98 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
47 \( ( 1 + 10 T + 50 T^{2} + 238 T^{3} + 1887 T^{4} + 22532 T^{5} + 159292 T^{6} + 22532 p T^{7} + 1887 p^{2} T^{8} + 238 p^{3} T^{9} + 50 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
53 \( 1 - 770 T^{4} - 973217 T^{8} + 40709312692 T^{12} - 973217 p^{4} T^{16} - 770 p^{8} T^{20} + p^{12} T^{24} \)
59 \( ( 1 - 94 T^{2} + 10295 T^{4} - 538692 T^{6} + 10295 p^{2} T^{8} - 94 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
61 \( ( 1 - 4 T + 179 T^{2} - 468 T^{3} + 179 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
67 \( 1 - 1522 T^{4} - 18418945 T^{8} + 74757937812 T^{12} - 18418945 p^{4} T^{16} - 1522 p^{8} T^{20} + p^{12} T^{24} \)
71 \( ( 1 - 322 T^{2} + 47167 T^{4} - 4165180 T^{6} + 47167 p^{2} T^{8} - 322 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
73 \( ( 1 - 38 T + 722 T^{2} - 10630 T^{3} + 136047 T^{4} - 1439940 T^{5} + 12990236 T^{6} - 1439940 p T^{7} + 136047 p^{2} T^{8} - 10630 p^{3} T^{9} + 722 p^{4} T^{10} - 38 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
79 \( ( 1 + 290 T^{2} + 39503 T^{4} + 3599964 T^{6} + 39503 p^{2} T^{8} + 290 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
83 \( ( 1 + 42 T + 882 T^{2} + 14230 T^{3} + 197415 T^{4} + 2254052 T^{5} + 21796604 T^{6} + 2254052 p T^{7} + 197415 p^{2} T^{8} + 14230 p^{3} T^{9} + 882 p^{4} T^{10} + 42 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
89 \( ( 1 + 246 T^{2} + 35247 T^{4} + 3456884 T^{6} + 35247 p^{2} T^{8} + 246 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
97 \( 1 - 40146 T^{4} + 781381263 T^{8} - 9244629922156 T^{12} + 781381263 p^{4} T^{16} - 40146 p^{8} T^{20} + p^{12} T^{24} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.79192991633008723189216285239, −2.75674818325843326679430523100, −2.66042632193977265496396930178, −2.64933427892789962311101790549, −2.60971551672145017262228683724, −2.45015442152031075964455844435, −2.37739957012908621033448102263, −2.26215148521439123216773014333, −2.02375350641487194391836885208, −1.96105696225305635015702470470, −1.94349557876035839662169115803, −1.88769515930737701205905740074, −1.57766763939987853950963319573, −1.53629259503570844216184496940, −1.51041281053989856775555401541, −1.33473805628746525279361945343, −1.31032539207231209814891767580, −1.23543733161376116338255634726, −0.982602516569116156717912729928, −0.885876502953331651791581248510, −0.841961934614638315574179660472, −0.51832180000096812932993913772, −0.42801790920419676168375386378, −0.42752399962368283076900292641, −0.17042453017812554007639641078, 0.17042453017812554007639641078, 0.42752399962368283076900292641, 0.42801790920419676168375386378, 0.51832180000096812932993913772, 0.841961934614638315574179660472, 0.885876502953331651791581248510, 0.982602516569116156717912729928, 1.23543733161376116338255634726, 1.31032539207231209814891767580, 1.33473805628746525279361945343, 1.51041281053989856775555401541, 1.53629259503570844216184496940, 1.57766763939987853950963319573, 1.88769515930737701205905740074, 1.94349557876035839662169115803, 1.96105696225305635015702470470, 2.02375350641487194391836885208, 2.26215148521439123216773014333, 2.37739957012908621033448102263, 2.45015442152031075964455844435, 2.60971551672145017262228683724, 2.64933427892789962311101790549, 2.66042632193977265496396930178, 2.75674818325843326679430523100, 2.79192991633008723189216285239

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

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