Properties

Label 24-12e36-1.1-c2e12-0-2
Degree $24$
Conductor $7.088\times 10^{38}$
Sign $1$
Analytic cond. $1.18725\times 10^{20}$
Root an. cond. $6.86182$
Motivic weight $2$
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  − 48·17-s + 114·25-s − 336·41-s + 318·49-s + 60·73-s − 1.24e3·89-s − 204·97-s − 1.20e3·113-s − 618·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 156·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 2.82·17-s + 4.55·25-s − 8.19·41-s + 6.48·49-s + 0.821·73-s − 14.0·89-s − 2.10·97-s − 10.6·113-s − 5.10·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.923·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯

Functional equation

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

Invariants

Degree: \(24\)
Conductor: \(2^{72} \cdot 3^{36}\)
Sign: $1$
Analytic conductor: \(1.18725\times 10^{20}\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{72} \cdot 3^{36} ,\ ( \ : [1]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.009720999027\)
\(L(\frac12)\) \(\approx\) \(0.009720999027\)
\(L(2)\) 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 \( ( 1 - 57 T^{2} + 1794 T^{4} - 41897 T^{6} + 1794 p^{4} T^{8} - 57 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
7 \( ( 1 - 159 T^{2} + 14466 T^{4} - 864799 T^{6} + 14466 p^{4} T^{8} - 159 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
11 \( ( 1 + 309 T^{2} + 40998 T^{4} + 4295297 T^{6} + 40998 p^{4} T^{8} + 309 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
13 \( ( 1 - 6 p T^{2} + 87 p^{2} T^{4} - 9725348 T^{6} + 87 p^{6} T^{8} - 6 p^{9} T^{10} + p^{12} T^{12} )^{2} \)
17 \( ( 1 + 12 T + 327 T^{2} + 24 T^{3} + 327 p^{2} T^{4} + 12 p^{4} T^{5} + p^{6} T^{6} )^{4} \)
19 \( ( 1 + 510 T^{2} + 235311 T^{4} + 79937732 T^{6} + 235311 p^{4} T^{8} + 510 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
23 \( ( 1 - 654 T^{2} + 784671 T^{4} - 690596 p^{2} T^{6} + 784671 p^{4} T^{8} - 654 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
29 \( ( 1 - 3534 T^{2} + 5907759 T^{4} - 6107279780 T^{6} + 5907759 p^{4} T^{8} - 3534 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
31 \( ( 1 - 2319 T^{2} + 3460386 T^{4} - 3880559887 T^{6} + 3460386 p^{4} T^{8} - 2319 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
37 \( ( 1 - 2958 T^{2} + 6236943 T^{4} - 8446524068 T^{6} + 6236943 p^{4} T^{8} - 2958 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
41 \( ( 1 + 84 T + 4071 T^{2} + 169224 T^{3} + 4071 p^{2} T^{4} + 84 p^{4} T^{5} + p^{6} T^{6} )^{4} \)
43 \( ( 1 + 6774 T^{2} + 22503903 T^{4} + 49221757172 T^{6} + 22503903 p^{4} T^{8} + 6774 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
47 \( ( 1 - 5766 T^{2} + 22375887 T^{4} - 58361753108 T^{6} + 22375887 p^{4} T^{8} - 5766 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
53 \( ( 1 - 15753 T^{2} + 106280994 T^{4} - 392809110617 T^{6} + 106280994 p^{4} T^{8} - 15753 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
59 \( ( 1 + 9654 T^{2} + 47947551 T^{4} + 179050130804 T^{6} + 47947551 p^{4} T^{8} + 9654 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
61 \( ( 1 - 6674 T^{2} + p^{4} T^{4} )^{6} \)
67 \( ( 1 + 20598 T^{2} + 198533823 T^{4} + 1132271795828 T^{6} + 198533823 p^{4} T^{8} + 20598 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
71 \( ( 1 - 7854 T^{2} + 6574047 T^{4} + 80774387548 T^{6} + 6574047 p^{4} T^{8} - 7854 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
73 \( ( 1 - 15 T + 10578 T^{2} - 280591 T^{3} + 10578 p^{2} T^{4} - 15 p^{4} T^{5} + p^{6} T^{6} )^{4} \)
79 \( ( 1 - 28578 T^{2} + 376929087 T^{4} - 474751804 p^{2} T^{6} + 376929087 p^{4} T^{8} - 28578 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
83 \( ( 1 + 10533 T^{2} - 37222170 T^{4} - 869901312751 T^{6} - 37222170 p^{4} T^{8} + 10533 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
89 \( ( 1 + 312 T + 52707 T^{2} + 5623536 T^{3} + 52707 p^{2} T^{4} + 312 p^{4} T^{5} + p^{6} T^{6} )^{4} \)
97 \( ( 1 + 51 T + 5862 T^{2} - 251561 T^{3} + 5862 p^{2} T^{4} + 51 p^{4} T^{5} + p^{6} T^{6} )^{4} \)
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

−2.65085370093428803966547081304, −2.62449936769427796470074675437, −2.59169869129692186636556756691, −2.57479472643802334869460541131, −2.45636774174422485230372817242, −2.30092928812064174152568026588, −2.24131876266110515994302927190, −2.13000329696054904903456383696, −1.80829386144890874222903171756, −1.74653184775907711815901396829, −1.62713142439393031249718112998, −1.49394068675048619738666217254, −1.39492309178523444560823938704, −1.36313940661806695158444562498, −1.30807121282824416229010146829, −1.27282481435830116132023301370, −1.16733195920597417685897709848, −1.11181626507160384118888392211, −1.10946425705734464540746263077, −0.70787331457370526183781694556, −0.49225840686527065640907329717, −0.30218725626771193379447580027, −0.18509257765859537000420117842, −0.05204167244406147679628815725, −0.04670289179555109894906296833, 0.04670289179555109894906296833, 0.05204167244406147679628815725, 0.18509257765859537000420117842, 0.30218725626771193379447580027, 0.49225840686527065640907329717, 0.70787331457370526183781694556, 1.10946425705734464540746263077, 1.11181626507160384118888392211, 1.16733195920597417685897709848, 1.27282481435830116132023301370, 1.30807121282824416229010146829, 1.36313940661806695158444562498, 1.39492309178523444560823938704, 1.49394068675048619738666217254, 1.62713142439393031249718112998, 1.74653184775907711815901396829, 1.80829386144890874222903171756, 2.13000329696054904903456383696, 2.24131876266110515994302927190, 2.30092928812064174152568026588, 2.45636774174422485230372817242, 2.57479472643802334869460541131, 2.59169869129692186636556756691, 2.62449936769427796470074675437, 2.65085370093428803966547081304

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

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