# Properties

 Label 10-8280e5-1.1-c1e5-0-0 Degree $10$ Conductor $3.892\times 10^{19}$ Sign $1$ Analytic cond. $1.26338\times 10^{9}$ Root an. cond. $8.13118$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 5·5-s − 4·7-s + 4·11-s + 4·13-s − 10·17-s − 4·19-s + 5·23-s + 15·25-s − 10·29-s + 6·31-s + 20·35-s + 6·37-s − 12·41-s − 2·43-s − 4·47-s − 20·55-s − 6·59-s + 16·61-s − 20·65-s − 4·67-s − 8·71-s + 14·73-s − 16·77-s + 18·79-s + 20·83-s + 50·85-s − 18·89-s + ⋯
 L(s)  = 1 − 2.23·5-s − 1.51·7-s + 1.20·11-s + 1.10·13-s − 2.42·17-s − 0.917·19-s + 1.04·23-s + 3·25-s − 1.85·29-s + 1.07·31-s + 3.38·35-s + 0.986·37-s − 1.87·41-s − 0.304·43-s − 0.583·47-s − 2.69·55-s − 0.781·59-s + 2.04·61-s − 2.48·65-s − 0.488·67-s − 0.949·71-s + 1.63·73-s − 1.82·77-s + 2.02·79-s + 2.19·83-s + 5.42·85-s − 1.90·89-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{10} \cdot 5^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{10} \cdot 5^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$10$$ Conductor: $$2^{15} \cdot 3^{10} \cdot 5^{5} \cdot 23^{5}$$ Sign: $1$ Analytic conductor: $$1.26338\times 10^{9}$$ Root analytic conductor: $$8.13118$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(10,\ 2^{15} \cdot 3^{10} \cdot 5^{5} \cdot 23^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.778262935$$ $$L(\frac12)$$ $$\approx$$ $$1.778262935$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
3 $$1$$
5$C_1$ $$( 1 + T )^{5}$$
23$C_1$ $$( 1 - T )^{5}$$
good7$C_2 \wr S_5$ $$1 + 4 T + 16 T^{2} + 58 T^{3} + 195 T^{4} + 548 T^{5} + 195 p T^{6} + 58 p^{2} T^{7} + 16 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10}$$
11$C_2 \wr S_5$ $$1 - 4 T + 9 T^{2} - 16 T^{3} + 100 T^{4} - 696 T^{5} + 100 p T^{6} - 16 p^{2} T^{7} + 9 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10}$$
13$C_2 \wr S_5$ $$1 - 4 T - T^{2} + 40 T^{3} + 84 T^{4} - 1192 T^{5} + 84 p T^{6} + 40 p^{2} T^{7} - p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10}$$
17$C_2 \wr S_5$ $$1 + 10 T + 60 T^{2} + 96 T^{3} - 533 T^{4} - 4932 T^{5} - 533 p T^{6} + 96 p^{2} T^{7} + 60 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10}$$
19$C_2 \wr S_5$ $$1 + 4 T + 41 T^{2} + 184 T^{3} + 788 T^{4} + 4616 T^{5} + 788 p T^{6} + 184 p^{2} T^{7} + 41 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10}$$
29$C_2 \wr S_5$ $$1 + 10 T + 120 T^{2} + 688 T^{3} + 4759 T^{4} + 21660 T^{5} + 4759 p T^{6} + 688 p^{2} T^{7} + 120 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10}$$
31$C_2 \wr S_5$ $$1 - 6 T + 76 T^{2} - 176 T^{3} + 2391 T^{4} - 3476 T^{5} + 2391 p T^{6} - 176 p^{2} T^{7} + 76 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10}$$
37$C_2 \wr S_5$ $$1 - 6 T + 174 T^{2} - 812 T^{3} + 12513 T^{4} - 43772 T^{5} + 12513 p T^{6} - 812 p^{2} T^{7} + 174 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10}$$
41$C_2 \wr S_5$ $$1 + 12 T + 152 T^{2} + 958 T^{3} + 7095 T^{4} + 36300 T^{5} + 7095 p T^{6} + 958 p^{2} T^{7} + 152 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10}$$
43$C_2 \wr S_5$ $$1 + 2 T + 131 T^{2} + 304 T^{3} + 8998 T^{4} + 18876 T^{5} + 8998 p T^{6} + 304 p^{2} T^{7} + 131 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10}$$
47$C_2 \wr S_5$ $$1 + 4 T + 181 T^{2} + 632 T^{3} + 14732 T^{4} + 42248 T^{5} + 14732 p T^{6} + 632 p^{2} T^{7} + 181 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10}$$
53$C_2 \wr S_5$ $$1 + 124 T^{2} - 202 T^{3} + 9035 T^{4} - 19068 T^{5} + 9035 p T^{6} - 202 p^{2} T^{7} + 124 p^{3} T^{8} + p^{5} T^{10}$$
59$C_2 \wr S_5$ $$1 + 6 T + 26 T^{2} - 120 T^{3} + 1229 T^{4} - 2684 T^{5} + 1229 p T^{6} - 120 p^{2} T^{7} + 26 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10}$$
61$C_2 \wr S_5$ $$1 - 16 T + 179 T^{2} - 536 T^{3} - 3328 T^{4} + 79536 T^{5} - 3328 p T^{6} - 536 p^{2} T^{7} + 179 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10}$$
67$C_2 \wr S_5$ $$1 + 4 T + 212 T^{2} + 910 T^{3} + 24327 T^{4} + 79852 T^{5} + 24327 p T^{6} + 910 p^{2} T^{7} + 212 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10}$$
71$C_2 \wr S_5$ $$1 + 8 T + 290 T^{2} + 1790 T^{3} + 37565 T^{4} + 179988 T^{5} + 37565 p T^{6} + 1790 p^{2} T^{7} + 290 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10}$$
73$C_2 \wr S_5$ $$1 - 14 T + 323 T^{2} - 3772 T^{3} + 44180 T^{4} - 402156 T^{5} + 44180 p T^{6} - 3772 p^{2} T^{7} + 323 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10}$$
79$C_2 \wr S_5$ $$1 - 18 T + 307 T^{2} - 3704 T^{3} + 40786 T^{4} - 393324 T^{5} + 40786 p T^{6} - 3704 p^{2} T^{7} + 307 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10}$$
83$C_2 \wr S_5$ $$1 - 20 T + 510 T^{2} - 6226 T^{3} + 88161 T^{4} - 748900 T^{5} + 88161 p T^{6} - 6226 p^{2} T^{7} + 510 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10}$$
89$C_2 \wr S_5$ $$1 + 18 T + 325 T^{2} + 4856 T^{3} + 55106 T^{4} + 582540 T^{5} + 55106 p T^{6} + 4856 p^{2} T^{7} + 325 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10}$$
97$C_2 \wr S_5$ $$1 - 4 T + 121 T^{2} - 192 T^{3} + 1702 T^{4} + 24072 T^{5} + 1702 p T^{6} - 192 p^{2} T^{7} + 121 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−4.52336351759448119970158099903, −4.38429776250523491488629837582, −4.14261257908554026966187488337, −4.08703299869568898134269867216, −3.94977482558191168904860297231, −3.87482670820588808630018444073, −3.57400840190912942996989389609, −3.46263438954715637089913477911, −3.43201298988962921435146036445, −3.42308137265890646370267167357, −3.08949296168369230736063862162, −2.76802605723353127996773736381, −2.66586559047192184633676977326, −2.62921771527450754792215467096, −2.56848309304838944365447128978, −1.90718926491961647063569073258, −1.89097787696178421380325412060, −1.87224405904429818286111278500, −1.54412907965128953180658996791, −1.52648971158569524364085052799, −0.958587136767787327850981862237, −0.72064640569503056221656053999, −0.58641742493072488737702014572, −0.45927436485173521828108752384, −0.21696996601724194680165685759, 0.21696996601724194680165685759, 0.45927436485173521828108752384, 0.58641742493072488737702014572, 0.72064640569503056221656053999, 0.958587136767787327850981862237, 1.52648971158569524364085052799, 1.54412907965128953180658996791, 1.87224405904429818286111278500, 1.89097787696178421380325412060, 1.90718926491961647063569073258, 2.56848309304838944365447128978, 2.62921771527450754792215467096, 2.66586559047192184633676977326, 2.76802605723353127996773736381, 3.08949296168369230736063862162, 3.42308137265890646370267167357, 3.43201298988962921435146036445, 3.46263438954715637089913477911, 3.57400840190912942996989389609, 3.87482670820588808630018444073, 3.94977482558191168904860297231, 4.08703299869568898134269867216, 4.14261257908554026966187488337, 4.38429776250523491488629837582, 4.52336351759448119970158099903