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Properties

Label 12-675e6-1.1-c1e6-0-1
Degree $12$
Conductor $9.459\times 10^{16}$
Sign $1$
Analytic cond. $24518.0$
Root an. cond. $2.32161$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s + 18·4-s + 6·7-s + 36·8-s − 18·11-s + 36·14-s + 54·16-s + 6·17-s − 12·19-s − 108·22-s + 18·23-s + 9·27-s + 108·28-s − 21·29-s − 3·31-s + 69·32-s + 36·34-s + 6·37-s − 72·38-s − 18·41-s − 6·43-s − 324·44-s + 108·46-s − 6·47-s + 12·49-s + 48·53-s + 54·54-s + ⋯
L(s)  = 1  + 4.24·2-s + 9·4-s + 2.26·7-s + 12.7·8-s − 5.42·11-s + 9.62·14-s + 27/2·16-s + 1.45·17-s − 2.75·19-s − 23.0·22-s + 3.75·23-s + 1.73·27-s + 20.4·28-s − 3.89·29-s − 0.538·31-s + 12.1·32-s + 6.17·34-s + 0.986·37-s − 11.6·38-s − 2.81·41-s − 0.914·43-s − 48.8·44-s + 15.9·46-s − 0.875·47-s + 12/7·49-s + 6.59·53-s + 7.34·54-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{18} \cdot 5^{12}\)
Sign: $1$
Analytic conductor: \(24518.0\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{18} \cdot 5^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(43.89641858\)
\(L(\frac12)\) \(\approx\) \(43.89641858\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - p^{2} T^{3} + p^{3} T^{6} \)
5 \( 1 \)
good2 \( 1 - 3 p T + 9 p T^{2} - 9 p^{2} T^{3} + 27 p T^{4} - 69 T^{5} + 91 T^{6} - 69 p T^{7} + 27 p^{3} T^{8} - 9 p^{5} T^{9} + 9 p^{5} T^{10} - 3 p^{6} T^{11} + p^{6} T^{12} \)
7 \( 1 - 6 T + 24 T^{2} - 64 T^{3} + 108 T^{4} - 108 T^{5} - 27 T^{6} - 108 p T^{7} + 108 p^{2} T^{8} - 64 p^{3} T^{9} + 24 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 18 T + 135 T^{2} + 513 T^{3} + 675 T^{4} - 3033 T^{5} - 18422 T^{6} - 3033 p T^{7} + 675 p^{2} T^{8} + 513 p^{3} T^{9} + 135 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 9 T^{2} - 43 T^{3} - 171 T^{4} + 639 T^{5} + 3702 T^{6} + 639 p T^{7} - 171 p^{2} T^{8} - 43 p^{3} T^{9} - 9 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 6 T - 24 T^{2} + 54 T^{3} + 1338 T^{4} - 1914 T^{5} - 18929 T^{6} - 1914 p T^{7} + 1338 p^{2} T^{8} + 54 p^{3} T^{9} - 24 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 12 T + 51 T^{2} + 188 T^{3} + 1314 T^{4} + 5076 T^{5} + 12699 T^{6} + 5076 p T^{7} + 1314 p^{2} T^{8} + 188 p^{3} T^{9} + 51 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 18 T + 198 T^{2} - 72 p T^{3} + 11520 T^{4} - 67716 T^{5} + 345961 T^{6} - 67716 p T^{7} + 11520 p^{2} T^{8} - 72 p^{4} T^{9} + 198 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 21 T + 207 T^{2} + 1125 T^{3} + 1890 T^{4} - 22776 T^{5} - 210023 T^{6} - 22776 p T^{7} + 1890 p^{2} T^{8} + 1125 p^{3} T^{9} + 207 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 3 T + 24 T^{2} + 80 T^{3} + 423 T^{4} - 4653 T^{5} - 5787 T^{6} - 4653 p T^{7} + 423 p^{2} T^{8} + 80 p^{3} T^{9} + 24 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 6 T - 48 T^{2} + 206 T^{3} + 1818 T^{4} - 738 T^{5} - 83025 T^{6} - 738 p T^{7} + 1818 p^{2} T^{8} + 206 p^{3} T^{9} - 48 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 18 T + 171 T^{2} + 1143 T^{3} + 9009 T^{4} + 79119 T^{5} + 597574 T^{6} + 79119 p T^{7} + 9009 p^{2} T^{8} + 1143 p^{3} T^{9} + 171 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 6 T + 156 T^{2} + 905 T^{3} + 13329 T^{4} + 66195 T^{5} + 704013 T^{6} + 66195 p T^{7} + 13329 p^{2} T^{8} + 905 p^{3} T^{9} + 156 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 6 T - 621 T^{3} - 2439 T^{4} + 6873 T^{5} + 245629 T^{6} + 6873 p T^{7} - 2439 p^{2} T^{8} - 621 p^{3} T^{9} + 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 - 24 T + 330 T^{2} - 2871 T^{3} + 330 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 144 T^{2} + 576 T^{3} + 144 p T^{4} + 97596 T^{5} + 406765 T^{6} + 97596 p T^{7} + 144 p^{3} T^{8} + 576 p^{3} T^{9} + 144 p^{4} T^{10} + p^{6} T^{12} \)
61 \( 1 - 36 T + 756 T^{2} - 190 p T^{3} + 141336 T^{4} - 1424232 T^{5} + 12082719 T^{6} - 1424232 p T^{7} + 141336 p^{2} T^{8} - 190 p^{4} T^{9} + 756 p^{4} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 18 T + 234 T^{2} - 2860 T^{3} + 30024 T^{4} - 262980 T^{5} + 2254197 T^{6} - 262980 p T^{7} + 30024 p^{2} T^{8} - 2860 p^{3} T^{9} + 234 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 3 T - 150 T^{2} - 63 T^{3} + 13371 T^{4} + 19698 T^{5} - 1081649 T^{6} + 19698 p T^{7} + 13371 p^{2} T^{8} - 63 p^{3} T^{9} - 150 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 6 T - 186 T^{2} - 418 T^{3} + 27828 T^{4} + 31752 T^{5} - 2245917 T^{6} + 31752 p T^{7} + 27828 p^{2} T^{8} - 418 p^{3} T^{9} - 186 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 12 T + 204 T^{2} + 2915 T^{3} + 31545 T^{4} + 326493 T^{5} + 3330513 T^{6} + 326493 p T^{7} + 31545 p^{2} T^{8} + 2915 p^{3} T^{9} + 204 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 18 T - 72 T^{2} + 3609 T^{3} - 20907 T^{4} - 178965 T^{5} + 3381445 T^{6} - 178965 p T^{7} - 20907 p^{2} T^{8} + 3609 p^{3} T^{9} - 72 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 3 T - 186 T^{2} + 585 T^{3} + 18915 T^{4} - 33708 T^{5} - 1703351 T^{6} - 33708 p T^{7} + 18915 p^{2} T^{8} + 585 p^{3} T^{9} - 186 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 3 T - 42 T^{2} + 1904 T^{3} - 7479 T^{4} - 60777 T^{5} + 2561289 T^{6} - 60777 p T^{7} - 7479 p^{2} T^{8} + 1904 p^{3} T^{9} - 42 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.35485562563716158603113467226, −5.30966986742205641358240483983, −5.14516272183433376460249677986, −5.02041862030733594152870550675, −5.01371616124218810612299370834, −4.94934747095947924249752960568, −4.70309415865211164949791064397, −4.57355598820358008491044977073, −4.35337154258135707798419040943, −4.24213228907600621135750554879, −3.73206515307309049690443858974, −3.72529039646307801446107936313, −3.48430748220785210620397971537, −3.45829288065826710324307098404, −3.35693026671192425824738746930, −2.92924545213226361576612813480, −2.65024887684641983447561909864, −2.52839805020763235455903558379, −2.30829941676216497058240199911, −2.29294713796537206193624710606, −2.01852628781566488668008560491, −1.96743743901241239253370325181, −1.34166349132634338594485166629, −0.76317113316738554600419595630, −0.56867753597151268684574138849, 0.56867753597151268684574138849, 0.76317113316738554600419595630, 1.34166349132634338594485166629, 1.96743743901241239253370325181, 2.01852628781566488668008560491, 2.29294713796537206193624710606, 2.30829941676216497058240199911, 2.52839805020763235455903558379, 2.65024887684641983447561909864, 2.92924545213226361576612813480, 3.35693026671192425824738746930, 3.45829288065826710324307098404, 3.48430748220785210620397971537, 3.72529039646307801446107936313, 3.73206515307309049690443858974, 4.24213228907600621135750554879, 4.35337154258135707798419040943, 4.57355598820358008491044977073, 4.70309415865211164949791064397, 4.94934747095947924249752960568, 5.01371616124218810612299370834, 5.02041862030733594152870550675, 5.14516272183433376460249677986, 5.30966986742205641358240483983, 5.35485562563716158603113467226

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

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