Properties

 Label 12-95e12-1.1-c1e6-0-8 Degree $12$ Conductor $5.404\times 10^{23}$ Sign $1$ Analytic cond. $1.40070\times 10^{11}$ Root an. cond. $8.48910$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $6$

Origins of factors

Dirichlet series

 L(s)  = 1 − 2·4-s − 2·9-s + 2·11-s − 3·16-s − 36·29-s + 4·36-s − 12·41-s − 4·44-s − 23·49-s − 20·59-s − 14·61-s + 8·64-s − 52·71-s + 24·79-s + 81-s − 24·89-s − 4·99-s + 20·101-s − 40·109-s + 72·116-s − 31·121-s + 127-s + 131-s + 137-s + 139-s + 6·144-s + 149-s + ⋯
 L(s)  = 1 − 4-s − 2/3·9-s + 0.603·11-s − 3/4·16-s − 6.68·29-s + 2/3·36-s − 1.87·41-s − 0.603·44-s − 3.28·49-s − 2.60·59-s − 1.79·61-s + 64-s − 6.17·71-s + 2.70·79-s + 1/9·81-s − 2.54·89-s − 0.402·99-s + 1.99·101-s − 3.83·109-s + 6.68·116-s − 2.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/2·144-s + 0.0819·149-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

 Degree: $$12$$ Conductor: $$5^{12} \cdot 19^{12}$$ Sign: $1$ Analytic conductor: $$1.40070\times 10^{11}$$ Root analytic conductor: $$8.48910$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{9025} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$6$$ Selberg data: $$(12,\ 5^{12} \cdot 19^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )$$

Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
19 $$1$$
good2 $$1 + p T^{2} + 7 T^{4} + 3 p^{2} T^{6} + 7 p^{2} T^{8} + p^{5} T^{10} + p^{6} T^{12}$$
3 $$1 + 2 T^{2} + p T^{4} + 20 T^{6} + p^{3} T^{8} + 2 p^{4} T^{10} + p^{6} T^{12}$$
7 $$1 + 23 T^{2} + 307 T^{4} + 2586 T^{6} + 307 p^{2} T^{8} + 23 p^{4} T^{10} + p^{6} T^{12}$$
11 $$( 1 - T + 17 T^{2} - 10 T^{3} + 17 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2}$$
13 $$1 + 50 T^{2} + 1315 T^{4} + 21108 T^{6} + 1315 p^{2} T^{8} + 50 p^{4} T^{10} + p^{6} T^{12}$$
17 $$1 + 43 T^{2} + 1331 T^{4} + 25042 T^{6} + 1331 p^{2} T^{8} + 43 p^{4} T^{10} + p^{6} T^{12}$$
23 $$1 + 102 T^{2} + 4831 T^{4} + 138580 T^{6} + 4831 p^{2} T^{8} + 102 p^{4} T^{10} + p^{6} T^{12}$$
29 $$( 1 + 6 T + p T^{2} )^{6}$$
31 $$( 1 + 37 T^{2} - 128 T^{3} + 37 p T^{4} + p^{3} T^{6} )^{2}$$
37 $$1 + 166 T^{2} + 13011 T^{4} + 608316 T^{6} + 13011 p^{2} T^{8} + 166 p^{4} T^{10} + p^{6} T^{12}$$
41 $$( 1 + 6 T + 79 T^{2} + 516 T^{3} + 79 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
43 $$1 + 239 T^{2} + 24571 T^{4} + 1388154 T^{6} + 24571 p^{2} T^{8} + 239 p^{4} T^{10} + p^{6} T^{12}$$
47 $$1 + 95 T^{2} + 5443 T^{4} + 214314 T^{6} + 5443 p^{2} T^{8} + 95 p^{4} T^{10} + p^{6} T^{12}$$
53 $$1 + 162 T^{2} + 11539 T^{4} + 610708 T^{6} + 11539 p^{2} T^{8} + 162 p^{4} T^{10} + p^{6} T^{12}$$
59 $$( 1 + 10 T + 185 T^{2} + 1132 T^{3} + 185 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
61 $$( 1 + 7 T + 79 T^{2} + 78 T^{3} + 79 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
67 $$1 + 62 T^{2} + 4771 T^{4} + 199788 T^{6} + 4771 p^{2} T^{8} + 62 p^{4} T^{10} + p^{6} T^{12}$$
71 $$( 1 + 26 T + 413 T^{2} + 4124 T^{3} + 413 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
73 $$1 + 307 T^{2} + 43299 T^{4} + 3822498 T^{6} + 43299 p^{2} T^{8} + 307 p^{4} T^{10} + p^{6} T^{12}$$
79 $$( 1 - 12 T + 229 T^{2} - 1864 T^{3} + 229 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
83 $$1 + 270 T^{2} + 39367 T^{4} + 3817060 T^{6} + 39367 p^{2} T^{8} + 270 p^{4} T^{10} + p^{6} T^{12}$$
89 $$( 1 + 12 T - 17 T^{2} - 1320 T^{3} - 17 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
97 $$1 + 554 T^{2} + 130507 T^{4} + 16717956 T^{6} + 130507 p^{2} T^{8} + 554 p^{4} T^{10} + p^{6} T^{12}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

Imaginary part of the first few zeros on the critical line

−4.34547556220371665069794603131, −4.20887088825725285190421617303, −4.04621034445874081826162491062, −3.80530448652832527833496737768, −3.71289717065864067989591732494, −3.65651988562626513715911276347, −3.60956299357297339670202169495, −3.49354331073470483610783517566, −3.27419653689933536535042735090, −3.15244046009554650056232409292, −3.13158245632159288104202864521, −2.70261483847247809347164966736, −2.67392876795961646102982202852, −2.66862137425607554548374885270, −2.63422165133234498498773838216, −2.18771958761694984759879362610, −1.91699508230189307836811504340, −1.83574618663439828440225426167, −1.83024407632055343428133975340, −1.76783679801406828889504003103, −1.53590705808010721915145214042, −1.38415201317307058256531224235, −1.25627488498706123569511945575, −1.05233988037200921314418825223, −0.806976729640905341138530580520, 0, 0, 0, 0, 0, 0, 0.806976729640905341138530580520, 1.05233988037200921314418825223, 1.25627488498706123569511945575, 1.38415201317307058256531224235, 1.53590705808010721915145214042, 1.76783679801406828889504003103, 1.83024407632055343428133975340, 1.83574618663439828440225426167, 1.91699508230189307836811504340, 2.18771958761694984759879362610, 2.63422165133234498498773838216, 2.66862137425607554548374885270, 2.67392876795961646102982202852, 2.70261483847247809347164966736, 3.13158245632159288104202864521, 3.15244046009554650056232409292, 3.27419653689933536535042735090, 3.49354331073470483610783517566, 3.60956299357297339670202169495, 3.65651988562626513715911276347, 3.71289717065864067989591732494, 3.80530448652832527833496737768, 4.04621034445874081826162491062, 4.20887088825725285190421617303, 4.34547556220371665069794603131

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

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