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Properties

Label 12-675e6-1.1-c1e6-0-1
Degree 1212
Conductor 9.459×10169.459\times 10^{16}
Sign 11
Analytic cond. 24518.024518.0
Root an. cond. 2.321612.32161
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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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

Λ(s)=((318⋅512)s/2 ΓC(s)6 L(s)=( Λ(2−s)\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}
Λ(s)=((318⋅512)s/2 ΓC(s+1/2)6 L(s)=( Λ(1−s)\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: 1212
Conductor: 318⋅5123^{18} \cdot 5^{12}
Sign: 11
Analytic conductor: 24518.024518.0
Root analytic conductor: 2.321612.32161
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 318⋅512, ( :[1/2]6), 1)(12,\ 3^{18} \cdot 5^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 43.8964185843.89641858
L(12)L(\frac12) ≈\approx 43.8964185843.89641858
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1−p2T3+p3T6 1 - p^{2} T^{3} + p^{3} T^{6}
5 1 1
good2 1−3pT+9pT2−9p2T3+27pT4−69T5+91T6−69pT7+27p3T8−9p5T9+9p5T10−3p6T11+p6T12 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 1−6T+24T2−64T3+108T4−108T5−27T6−108pT7+108p2T8−64p3T9+24p4T10−6p5T11+p6T12 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+18T+135T2+513T3+675T4−3033T5−18422T6−3033pT7+675p2T8+513p3T9+135p4T10+18p5T11+p6T12 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 1−9T2−43T3−171T4+639T5+3702T6+639pT7−171p2T8−43p3T9−9p4T10+p6T12 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 1−6T−24T2+54T3+1338T4−1914T5−18929T6−1914pT7+1338p2T8+54p3T9−24p4T10−6p5T11+p6T12 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+12T+51T2+188T3+1314T4+5076T5+12699T6+5076pT7+1314p2T8+188p3T9+51p4T10+12p5T11+p6T12 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 1−18T+198T2−72pT3+11520T4−67716T5+345961T6−67716pT7+11520p2T8−72p4T9+198p4T10−18p5T11+p6T12 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+21T+207T2+1125T3+1890T4−22776T5−210023T6−22776pT7+1890p2T8+1125p3T9+207p4T10+21p5T11+p6T12 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+3T+24T2+80T3+423T4−4653T5−5787T6−4653pT7+423p2T8+80p3T9+24p4T10+3p5T11+p6T12 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 1−6T−48T2+206T3+1818T4−738T5−83025T6−738pT7+1818p2T8+206p3T9−48p4T10−6p5T11+p6T12 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+18T+171T2+1143T3+9009T4+79119T5+597574T6+79119pT7+9009p2T8+1143p3T9+171p4T10+18p5T11+p6T12 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+6T+156T2+905T3+13329T4+66195T5+704013T6+66195pT7+13329p2T8+905p3T9+156p4T10+6p5T11+p6T12 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+6T−621T3−2439T4+6873T5+245629T6+6873pT7−2439p2T8−621p3T9+6p5T11+p6T12 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 (1−24T+330T2−2871T3+330pT4−24p2T5+p3T6)2 ( 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+144T2+576T3+144pT4+97596T5+406765T6+97596pT7+144p3T8+576p3T9+144p4T10+p6T12 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 1−36T+756T2−190pT3+141336T4−1424232T5+12082719T6−1424232pT7+141336p2T8−190p4T9+756p4T10−36p5T11+p6T12 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 1−18T+234T2−2860T3+30024T4−262980T5+2254197T6−262980pT7+30024p2T8−2860p3T9+234p4T10−18p5T11+p6T12 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 1−3T−150T2−63T3+13371T4+19698T5−1081649T6+19698pT7+13371p2T8−63p3T9−150p4T10−3p5T11+p6T12 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+6T−186T2−418T3+27828T4+31752T5−2245917T6+31752pT7+27828p2T8−418p3T9−186p4T10+6p5T11+p6T12 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+12T+204T2+2915T3+31545T4+326493T5+3330513T6+326493pT7+31545p2T8+2915p3T9+204p4T10+12p5T11+p6T12 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 1−18T−72T2+3609T3−20907T4−178965T5+3381445T6−178965pT7−20907p2T8+3609p3T9−72p4T10−18p5T11+p6T12 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 1−3T−186T2+585T3+18915T4−33708T5−1703351T6−33708pT7+18915p2T8+585p3T9−186p4T10−3p5T11+p6T12 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 1−3T−42T2+1904T3−7479T4−60777T5+2561289T6−60777pT7−7479p2T8+1904p3T9−42p4T10−3p5T11+p6T12 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)=∏p ∏j=112(1−αj,p p−s)−1L(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 ZZ-function along the critical line

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