# Properties

 Label 24-325e12-1.1-c1e12-0-1 Degree $24$ Conductor $1.389\times 10^{30}$ Sign $1$ Analytic cond. $93309.4$ Root an. cond. $1.61094$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·4-s + 6·9-s + 6·16-s − 12·19-s − 18·29-s − 16·31-s + 24·36-s + 14·41-s + 18·49-s + 4·59-s + 6·61-s − 6·64-s − 12·71-s − 48·76-s + 104·79-s + 26·81-s − 20·89-s − 26·101-s − 24·109-s − 72·116-s + 40·121-s − 64·124-s + 127-s + 131-s + 137-s + 139-s + 36·144-s + ⋯
 L(s)  = 1 + 2·4-s + 2·9-s + 3/2·16-s − 2.75·19-s − 3.34·29-s − 2.87·31-s + 4·36-s + 2.18·41-s + 18/7·49-s + 0.520·59-s + 0.768·61-s − 3/4·64-s − 1.42·71-s − 5.50·76-s + 11.7·79-s + 26/9·81-s − 2.11·89-s − 2.58·101-s − 2.29·109-s − 6.68·116-s + 3.63·121-s − 5.74·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3·144-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$24$$ Conductor: $$5^{24} \cdot 13^{12}$$ Sign: $1$ Analytic conductor: $$93309.4$$ Root analytic conductor: $$1.61094$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{325} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(24,\ 5^{24} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$5.712969930$$ $$L(\frac12)$$ $$\approx$$ $$5.712969930$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
13 $$1 + 15 T^{2} + 3 p T^{4} + 322 T^{6} + 3 p^{3} T^{8} + 15 p^{4} T^{10} + p^{6} T^{12}$$
good2 $$1 - p^{2} T^{2} + 5 p T^{4} - 5 p T^{6} - p^{4} T^{8} + 43 p T^{10} - 223 T^{12} + 43 p^{3} T^{14} - p^{8} T^{16} - 5 p^{7} T^{18} + 5 p^{9} T^{20} - p^{12} T^{22} + p^{12} T^{24}$$
3 $$1 - 2 p T^{2} + 10 T^{4} + 40 T^{6} - 146 T^{8} - 226 T^{10} + 2062 T^{12} - 226 p^{2} T^{14} - 146 p^{4} T^{16} + 40 p^{6} T^{18} + 10 p^{8} T^{20} - 2 p^{11} T^{22} + p^{12} T^{24}$$
7 $$1 - 18 T^{2} + 82 T^{4} - 536 T^{6} + 12326 T^{8} - 78550 T^{10} + 235838 T^{12} - 78550 p^{2} T^{14} + 12326 p^{4} T^{16} - 536 p^{6} T^{18} + 82 p^{8} T^{20} - 18 p^{10} T^{22} + p^{12} T^{24}$$
11 $$( 1 - 20 T^{2} - 16 T^{3} + 180 T^{4} + 160 T^{5} - 1674 T^{6} + 160 p T^{7} + 180 p^{2} T^{8} - 16 p^{3} T^{9} - 20 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
17 $$1 - 67 T^{2} + 2371 T^{4} - 56020 T^{6} + 996641 T^{8} - 14527513 T^{10} + 221018582 T^{12} - 14527513 p^{2} T^{14} + 996641 p^{4} T^{16} - 56020 p^{6} T^{18} + 2371 p^{8} T^{20} - 67 p^{10} T^{22} + p^{12} T^{24}$$
19 $$( 1 + 6 T - 20 T^{2} - 100 T^{3} + 764 T^{4} + 1606 T^{5} - 11338 T^{6} + 1606 p T^{7} + 764 p^{2} T^{8} - 100 p^{3} T^{9} - 20 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2}$$
23 $$1 - 126 T^{2} + 9010 T^{4} - 451400 T^{6} + 17445734 T^{8} - 540254266 T^{10} + 13705453502 T^{12} - 540254266 p^{2} T^{14} + 17445734 p^{4} T^{16} - 451400 p^{6} T^{18} + 9010 p^{8} T^{20} - 126 p^{10} T^{22} + p^{12} T^{24}$$
29 $$( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{6}$$
31 $$( 1 + 4 T + 53 T^{2} + 288 T^{3} + 53 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{4}$$
37 $$1 - 187 T^{2} + 19451 T^{4} - 1426940 T^{6} + 81658601 T^{8} - 3849109833 T^{10} + 153778997622 T^{12} - 3849109833 p^{2} T^{14} + 81658601 p^{4} T^{16} - 1426940 p^{6} T^{18} + 19451 p^{8} T^{20} - 187 p^{10} T^{22} + p^{12} T^{24}$$
41 $$( 1 - 7 T - 45 T^{2} + 500 T^{3} + 929 T^{4} - 12237 T^{5} + 28438 T^{6} - 12237 p T^{7} + 929 p^{2} T^{8} + 500 p^{3} T^{9} - 45 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} )^{2}$$
43 $$1 - 178 T^{2} + 17426 T^{4} - 1084520 T^{6} + 47574566 T^{8} - 1584620742 T^{10} + 56532979182 T^{12} - 1584620742 p^{2} T^{14} + 47574566 p^{4} T^{16} - 1084520 p^{6} T^{18} + 17426 p^{8} T^{20} - 178 p^{10} T^{22} + p^{12} T^{24}$$
47 $$( 1 + 46 T^{2} + 3407 T^{4} + 54276 T^{6} + 3407 p^{2} T^{8} + 46 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
53 $$( 1 + 147 T^{2} + 6923 T^{4} + 205346 T^{6} + 6923 p^{2} T^{8} + 147 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
59 $$( 1 - 2 T - 2 p T^{2} - 44 T^{3} + 7486 T^{4} + 9630 T^{5} - 471322 T^{6} + 9630 p T^{7} + 7486 p^{2} T^{8} - 44 p^{3} T^{9} - 2 p^{5} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2}$$
61 $$( 1 - 3 T - 125 T^{2} + 100 T^{3} + 9029 T^{4} + 4247 T^{5} - 611842 T^{6} + 4247 p T^{7} + 9029 p^{2} T^{8} + 100 p^{3} T^{9} - 125 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2}$$
67 $$1 - 302 T^{2} + 48066 T^{4} - 5726680 T^{6} + 564790166 T^{8} - 46598346138 T^{10} + 3315667632142 T^{12} - 46598346138 p^{2} T^{14} + 564790166 p^{4} T^{16} - 5726680 p^{6} T^{18} + 48066 p^{8} T^{20} - 302 p^{10} T^{22} + p^{12} T^{24}$$
71 $$( 1 + 6 T - 176 T^{2} - 380 T^{3} + 24936 T^{4} + 24734 T^{5} - 1974402 T^{6} + 24734 p T^{7} + 24936 p^{2} T^{8} - 380 p^{3} T^{9} - 176 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2}$$
73 $$( 1 + 223 T^{2} + 31055 T^{4} + 2685330 T^{6} + 31055 p^{2} T^{8} + 223 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
79 $$( 1 - 26 T + 417 T^{2} - 4268 T^{3} + 417 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} )^{4}$$
83 $$( 1 + 222 T^{2} + 35303 T^{4} + 3305156 T^{6} + 35303 p^{2} T^{8} + 222 p^{4} T^{10} + p^{6} T^{12} )^{2}$$
89 $$( 1 + 10 T - 10 T^{2} + 8 p T^{3} + 370 T^{4} - 95250 T^{5} - 421426 T^{6} - 95250 p T^{7} + 370 p^{2} T^{8} + 8 p^{4} T^{9} - 10 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} )^{2}$$
97 $$1 - 302 T^{2} + 44346 T^{4} - 4094200 T^{6} + 2434958 p T^{8} + 3936373062 T^{10} - 1781970067778 T^{12} + 3936373062 p^{2} T^{14} + 2434958 p^{5} T^{16} - 4094200 p^{6} T^{18} + 44346 p^{8} T^{20} - 302 p^{10} T^{22} + p^{12} T^{24}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$