# Properties

 Label 24-3e72-1.1-c1e12-0-15 Degree $24$ Conductor $2.253\times 10^{34}$ Sign $1$ Analytic cond. $1.51375\times 10^{9}$ Root an. cond. $2.41269$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 6·4-s + 12·7-s + 12·13-s + 21·16-s − 48·19-s + 15·25-s + 72·28-s + 30·31-s − 12·37-s + 12·43-s + 96·49-s + 72·52-s + 12·61-s + 64·64-s − 6·67-s + 24·73-s − 288·76-s + 48·79-s + 144·91-s + 12·97-s + 90·100-s + 12·103-s + 24·109-s + 252·112-s + 24·121-s + 180·124-s + 127-s + ⋯
 L(s)  = 1 + 3·4-s + 4.53·7-s + 3.32·13-s + 21/4·16-s − 11.0·19-s + 3·25-s + 13.6·28-s + 5.38·31-s − 1.97·37-s + 1.82·43-s + 96/7·49-s + 9.98·52-s + 1.53·61-s + 8·64-s − 0.733·67-s + 2.80·73-s − 33.0·76-s + 5.40·79-s + 15.0·91-s + 1.21·97-s + 9·100-s + 1.18·103-s + 2.29·109-s + 23.8·112-s + 2.18·121-s + 16.1·124-s + 0.0887·127-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$24$$ Conductor: $$3^{72}$$ Sign: $1$ Analytic conductor: $$1.51375\times 10^{9}$$ Root analytic conductor: $$2.41269$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(24,\ 3^{72} ,\ ( \ : [1/2]^{12} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$114.9665434$$ $$L(\frac12)$$ $$\approx$$ $$114.9665434$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
good2 $$1 - 3 p T^{2} + 15 T^{4} - 7 p^{2} T^{6} + 63 T^{8} - 117 T^{10} + 189 T^{12} - 117 p^{2} T^{14} + 63 p^{4} T^{16} - 7 p^{8} T^{18} + 15 p^{8} T^{20} - 3 p^{11} T^{22} + p^{12} T^{24}$$
5 $$1 - 3 p T^{2} + 132 T^{4} - 541 T^{6} - 81 T^{8} + 20664 T^{10} - 143271 T^{12} + 20664 p^{2} T^{14} - 81 p^{4} T^{16} - 541 p^{6} T^{18} + 132 p^{8} T^{20} - 3 p^{11} T^{22} + p^{12} T^{24}$$
7 $$( 1 - 6 T + 6 T^{2} - 10 T^{3} + 180 T^{4} - 324 T^{5} - 153 T^{6} - 324 p T^{7} + 180 p^{2} T^{8} - 10 p^{3} T^{9} + 6 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2}$$
11 $$1 - 24 T^{2} + 204 T^{4} - 1018 T^{6} - 1548 T^{8} + 337932 T^{10} - 5992533 T^{12} + 337932 p^{2} T^{14} - 1548 p^{4} T^{16} - 1018 p^{6} T^{18} + 204 p^{8} T^{20} - 24 p^{10} T^{22} + p^{12} T^{24}$$
13 $$( 1 - 6 T + 6 T^{2} - 10 T^{3} + 1116 T^{5} - 6165 T^{6} + 1116 p T^{7} - 10 p^{3} T^{9} + 6 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2}$$
17 $$( 1 + 21 T^{2} + 582 T^{4} + 7729 T^{6} + 582 p^{2} T^{8} + 21 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
19 $$( 1 + 12 T + 96 T^{2} + 25 p T^{3} + 96 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{4}$$
23 $$1 - 24 T^{2} - 120 T^{4} + 43478 T^{6} - 605952 T^{8} - 9986688 T^{10} + 784095579 T^{12} - 9986688 p^{2} T^{14} - 605952 p^{4} T^{16} + 43478 p^{6} T^{18} - 120 p^{8} T^{20} - 24 p^{10} T^{22} + p^{12} T^{24}$$
29 $$1 - 123 T^{2} + 7620 T^{4} - 376993 T^{6} + 16453251 T^{8} - 585750852 T^{10} + 17766055521 T^{12} - 585750852 p^{2} T^{14} + 16453251 p^{4} T^{16} - 376993 p^{6} T^{18} + 7620 p^{8} T^{20} - 123 p^{10} T^{22} + p^{12} T^{24}$$
31 $$( 1 - 15 T + 69 T^{2} - 334 T^{3} + 4455 T^{4} - 25803 T^{5} + 94950 T^{6} - 25803 p T^{7} + 4455 p^{2} T^{8} - 334 p^{3} T^{9} + 69 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} )^{2}$$
37 $$( 1 + 3 T + 105 T^{2} + 205 T^{3} + 105 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{4}$$
41 $$1 - 159 T^{2} + 12480 T^{4} - 746281 T^{6} + 39865419 T^{8} - 1857162780 T^{10} + 78034591881 T^{12} - 1857162780 p^{2} T^{14} + 39865419 p^{4} T^{16} - 746281 p^{6} T^{18} + 12480 p^{8} T^{20} - 159 p^{10} T^{22} + p^{12} T^{24}$$
43 $$( 1 - 6 T - 93 T^{2} + 242 T^{3} + 8046 T^{4} - 234 p T^{5} - 365049 T^{6} - 234 p^{2} T^{7} + 8046 p^{2} T^{8} + 242 p^{3} T^{9} - 93 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2}$$
47 $$1 - 267 T^{2} + 40920 T^{4} - 4343077 T^{6} + 352503999 T^{8} - 22655835000 T^{10} + 1181028706929 T^{12} - 22655835000 p^{2} T^{14} + 352503999 p^{4} T^{16} - 4343077 p^{6} T^{18} + 40920 p^{8} T^{20} - 267 p^{10} T^{22} + p^{12} T^{24}$$
53 $$( 1 + 210 T^{2} + 21831 T^{4} + 1416508 T^{6} + 21831 p^{2} T^{8} + 210 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
59 $$1 - 168 T^{2} + 18348 T^{4} - 846154 T^{6} - 5136156 T^{8} + 5500747836 T^{10} - 431912032389 T^{12} + 5500747836 p^{2} T^{14} - 5136156 p^{4} T^{16} - 846154 p^{6} T^{18} + 18348 p^{8} T^{20} - 168 p^{10} T^{22} + p^{12} T^{24}$$
61 $$( 1 - 6 T - 75 T^{2} + 206 T^{3} + 2934 T^{4} + 10530 T^{5} - 241479 T^{6} + 10530 p T^{7} + 2934 p^{2} T^{8} + 206 p^{3} T^{9} - 75 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2}$$
67 $$( 1 + 3 T - 48 T^{2} - 1135 T^{3} - 2529 T^{4} + 27774 T^{5} + 769851 T^{6} + 27774 p T^{7} - 2529 p^{2} T^{8} - 1135 p^{3} T^{9} - 48 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2}$$
71 $$( 1 + 354 T^{2} + 56463 T^{4} + 5162812 T^{6} + 56463 p^{2} T^{8} + 354 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
73 $$( 1 - 6 T + 150 T^{2} - 965 T^{3} + 150 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{4}$$
79 $$( 1 - 24 T + 204 T^{2} - 1558 T^{3} + 20556 T^{4} - 109980 T^{5} + 38151 T^{6} - 109980 p T^{7} + 20556 p^{2} T^{8} - 1558 p^{3} T^{9} + 204 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} )^{2}$$
83 $$1 - 60 T^{2} - 6096 T^{4} + 169334 T^{6} + 4683852 T^{8} + 1526686308 T^{10} - 23585193477 T^{12} + 1526686308 p^{2} T^{14} + 4683852 p^{4} T^{16} + 169334 p^{6} T^{18} - 6096 p^{8} T^{20} - 60 p^{10} T^{22} + p^{12} T^{24}$$
89 $$( 1 + 147 T^{2} - 2811 T^{4} - 1426997 T^{6} - 2811 p^{2} T^{8} + 147 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
97 $$( 1 - 6 T - 156 T^{2} - 118 T^{3} + 15138 T^{4} + 67770 T^{5} - 1821897 T^{6} + 67770 p T^{7} + 15138 p^{2} T^{8} - 118 p^{3} T^{9} - 156 p^{4} T^{10} - 6 p^{5} T^{11} + p^{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

−3.48835037150359703236603631012, −3.27315380605514617259578191652, −3.20090265300473896796910375043, −2.89137451273538796769237969400, −2.84274357397700214846368122985, −2.65756551495959916273412622332, −2.58991793415285694948798626767, −2.43773321636966622664414556401, −2.43306030462151300093697595388, −2.37131374196383314022431500510, −2.35938108734026969022796893829, −2.27576372019749432269423232845, −2.02600398039401250120128338179, −1.93743128401385330418786249270, −1.92977482336545821204068274227, −1.83876358888238986358815443859, −1.82545267961753784815488932767, −1.57585044773283256261531834619, −1.30813519505957701374671381589, −1.28206134066438379642344795401, −1.02574998402616349812769196243, −0.892610915951764119589964175890, −0.837996596324665215434696160837, −0.72657846915495905354737958519, −0.43283030403012660915918717608, 0.43283030403012660915918717608, 0.72657846915495905354737958519, 0.837996596324665215434696160837, 0.892610915951764119589964175890, 1.02574998402616349812769196243, 1.28206134066438379642344795401, 1.30813519505957701374671381589, 1.57585044773283256261531834619, 1.82545267961753784815488932767, 1.83876358888238986358815443859, 1.92977482336545821204068274227, 1.93743128401385330418786249270, 2.02600398039401250120128338179, 2.27576372019749432269423232845, 2.35938108734026969022796893829, 2.37131374196383314022431500510, 2.43306030462151300093697595388, 2.43773321636966622664414556401, 2.58991793415285694948798626767, 2.65756551495959916273412622332, 2.84274357397700214846368122985, 2.89137451273538796769237969400, 3.20090265300473896796910375043, 3.27315380605514617259578191652, 3.48835037150359703236603631012

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

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