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Properties

Label 16-42e8-1.1-c4e8-0-1
Degree 1616
Conductor 9.683×10129.683\times 10^{12}
Sign 11
Analytic cond. 126226.126226.
Root an. cond. 2.083632.08363
Motivic weight 44
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 12·3-s − 32·4-s + 148·9-s − 384·12-s − 520·13-s + 640·16-s + 1.33e3·19-s + 664·25-s + 1.21e3·27-s + 4.88e3·31-s − 4.73e3·36-s − 800·37-s − 6.24e3·39-s − 2.81e3·43-s + 7.68e3·48-s + 1.37e3·49-s + 1.66e4·52-s + 1.60e4·57-s + 2.69e3·61-s − 1.02e4·64-s − 6.96e3·67-s + 2.76e4·73-s + 7.96e3·75-s − 4.27e4·76-s − 1.02e3·79-s + 1.37e4·81-s + 5.85e4·93-s + ⋯
L(s)  = 1  + 4/3·3-s − 2·4-s + 1.82·9-s − 8/3·12-s − 3.07·13-s + 5/2·16-s + 3.70·19-s + 1.06·25-s + 1.66·27-s + 5.07·31-s − 3.65·36-s − 0.584·37-s − 4.10·39-s − 1.52·43-s + 10/3·48-s + 4/7·49-s + 6.15·52-s + 4.93·57-s + 0.724·61-s − 5/2·64-s − 1.55·67-s + 5.18·73-s + 1.41·75-s − 7.40·76-s − 0.164·79-s + 2.09·81-s + 6.77·93-s + ⋯

Functional equation

Λ(s)=((28⋅38⋅78)s/2 ΓC(s)8 L(s)=( Λ(5−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}
Λ(s)=((28⋅38⋅78)s/2 ΓC(s+2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 28⋅38⋅782^{8} \cdot 3^{8} \cdot 7^{8}
Sign: 11
Analytic conductor: 126226.126226.
Root analytic conductor: 2.083632.08363
Motivic weight: 44
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 28⋅38⋅78, ( :[2]8), 1)(16,\ 2^{8} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : [2]^{8} ),\ 1 )

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 5.5074374625.507437462
L(12)L(\frac12) ≈\approx 5.5074374625.507437462
L(3)L(3) 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)
bad2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
3 1−4pT−4T2+68p2T3−662p2T4+68p6T5−4p8T6−4p13T7+p16T8 1 - 4 p T - 4 T^{2} + 68 p^{2} T^{3} - 662 p^{2} T^{4} + 68 p^{6} T^{5} - 4 p^{8} T^{6} - 4 p^{13} T^{7} + p^{16} T^{8}
7 (1−p3T2)4 ( 1 - p^{3} T^{2} )^{4}
good5 1−664T2+437924T4−88479624T6+4286559494T8−88479624p8T10+437924p16T12−664p24T14+p32T16 1 - 664 T^{2} + 437924 T^{4} - 88479624 T^{6} + 4286559494 T^{8} - 88479624 p^{8} T^{10} + 437924 p^{16} T^{12} - 664 p^{24} T^{14} + p^{32} T^{16}
11 1−75544T2+2921639132T4−72856032921000T6+1264800259684876358T8−72856032921000p8T10+2921639132p16T12−75544p24T14+p32T16 1 - 75544 T^{2} + 2921639132 T^{4} - 72856032921000 T^{6} + 1264800259684876358 T^{8} - 72856032921000 p^{8} T^{10} + 2921639132 p^{16} T^{12} - 75544 p^{24} T^{14} + p^{32} T^{16}
13 (1+20pT+63068T2+7956276T3+1342851002T4+7956276p4T5+63068p8T6+20p13T7+p16T8)2 ( 1 + 20 p T + 63068 T^{2} + 7956276 T^{3} + 1342851002 T^{4} + 7956276 p^{4} T^{5} + 63068 p^{8} T^{6} + 20 p^{13} T^{7} + p^{16} T^{8} )^{2}
17 1−71224T2+9382405916T4−1222841841739656T6+85209970415957819462T8−1222841841739656p8T10+9382405916p16T12−71224p24T14+p32T16 1 - 71224 T^{2} + 9382405916 T^{4} - 1222841841739656 T^{6} + 85209970415957819462 T^{8} - 1222841841739656 p^{8} T^{10} + 9382405916 p^{16} T^{12} - 71224 p^{24} T^{14} + p^{32} T^{16}
19 (1−668T+562204T2−226309484T3+108623604730T4−226309484p4T5+562204p8T6−668p12T7+p16T8)2 ( 1 - 668 T + 562204 T^{2} - 226309484 T^{3} + 108623604730 T^{4} - 226309484 p^{4} T^{5} + 562204 p^{8} T^{6} - 668 p^{12} T^{7} + p^{16} T^{8} )^{2}
23 1−954952T2+314690473244T4−8418785376240120T6− 1 - 954952 T^{2} + 314690473244 T^{4} - 8418785376240120 T^{6} - 14 ⁣⋯ ⁣2214\!\cdots\!22T8−8418785376240120p8T10+314690473244p16T12−954952p24T14+p32T16 T^{8} - 8418785376240120 p^{8} T^{10} + 314690473244 p^{16} T^{12} - 954952 p^{24} T^{14} + p^{32} T^{16}
29 1−4654456T2+10061048290844T4−13139168714386018632T6+ 1 - 4654456 T^{2} + 10061048290844 T^{4} - 13139168714386018632 T^{6} + 11 ⁣⋯ ⁣4211\!\cdots\!42T8−13139168714386018632p8T10+10061048290844p16T12−4654456p24T14+p32T16 T^{8} - 13139168714386018632 p^{8} T^{10} + 10061048290844 p^{16} T^{12} - 4654456 p^{24} T^{14} + p^{32} T^{16}
31 (1−2440T+4750052T2−5895563736T3+6660667523078T4−5895563736p4T5+4750052p8T6−2440p12T7+p16T8)2 ( 1 - 2440 T + 4750052 T^{2} - 5895563736 T^{3} + 6660667523078 T^{4} - 5895563736 p^{4} T^{5} + 4750052 p^{8} T^{6} - 2440 p^{12} T^{7} + p^{16} T^{8} )^{2}
37 (1+400T+2084716T2+395204848T3+5059454317606T4+395204848p4T5+2084716p8T6+400p12T7+p16T8)2 ( 1 + 400 T + 2084716 T^{2} + 395204848 T^{3} + 5059454317606 T^{4} + 395204848 p^{4} T^{5} + 2084716 p^{8} T^{6} + 400 p^{12} T^{7} + p^{16} T^{8} )^{2}
41 1−19751576T2+177774313337692T4− 1 - 19751576 T^{2} + 177774313337692 T^{4} - 95 ⁣⋯ ⁣1295\!\cdots\!12T6+ T^{6} + 32 ⁣⋯ ⁣9432\!\cdots\!94T8− T^{8} - 95 ⁣⋯ ⁣1295\!\cdots\!12p8T10+177774313337692p16T12−19751576p24T14+p32T16 p^{8} T^{10} + 177774313337692 p^{16} T^{12} - 19751576 p^{24} T^{14} + p^{32} T^{16}
43 (1+1408T+10046860T2+11972332672T3+48969670597606T4+11972332672p4T5+10046860p8T6+1408p12T7+p16T8)2 ( 1 + 1408 T + 10046860 T^{2} + 11972332672 T^{3} + 48969670597606 T^{4} + 11972332672 p^{4} T^{5} + 10046860 p^{8} T^{6} + 1408 p^{12} T^{7} + p^{16} T^{8} )^{2}
47 1−24321800T2+280807668157468T4− 1 - 24321800 T^{2} + 280807668157468 T^{4} - 20 ⁣⋯ ⁣2020\!\cdots\!20T6+ T^{6} + 11 ⁣⋯ ⁣6611\!\cdots\!66T8− T^{8} - 20 ⁣⋯ ⁣2020\!\cdots\!20p8T10+280807668157468p16T12−24321800p24T14+p32T16 p^{8} T^{10} + 280807668157468 p^{16} T^{12} - 24321800 p^{24} T^{14} + p^{32} T^{16}
53 1−46463768T2+1024233601864924T4− 1 - 46463768 T^{2} + 1024233601864924 T^{4} - 14 ⁣⋯ ⁣9614\!\cdots\!96T6+ T^{6} + 13 ⁣⋯ ⁣9413\!\cdots\!94T8− T^{8} - 14 ⁣⋯ ⁣9614\!\cdots\!96p8T10+1024233601864924p16T12−46463768p24T14+p32T16 p^{8} T^{10} + 1024233601864924 p^{16} T^{12} - 46463768 p^{24} T^{14} + p^{32} T^{16}
59 1+10637576T2+84111045669860T4+49194962777684533080T6− 1 + 10637576 T^{2} + 84111045669860 T^{4} + 49194962777684533080 T^{6} - 15 ⁣⋯ ⁣7815\!\cdots\!78T8+49194962777684533080p8T10+84111045669860p16T12+10637576p24T14+p32T16 T^{8} + 49194962777684533080 p^{8} T^{10} + 84111045669860 p^{16} T^{12} + 10637576 p^{24} T^{14} + p^{32} T^{16}
61 (1−1348T+29646524T2−20047475892T3+480742537431098T4−20047475892p4T5+29646524p8T6−1348p12T7+p16T8)2 ( 1 - 1348 T + 29646524 T^{2} - 20047475892 T^{3} + 480742537431098 T^{4} - 20047475892 p^{4} T^{5} + 29646524 p^{8} T^{6} - 1348 p^{12} T^{7} + p^{16} T^{8} )^{2}
67 (1+3480T+50068124T2+145959249576T3+1178892510670470T4+145959249576p4T5+50068124p8T6+3480p12T7+p16T8)2 ( 1 + 3480 T + 50068124 T^{2} + 145959249576 T^{3} + 1178892510670470 T^{4} + 145959249576 p^{4} T^{5} + 50068124 p^{8} T^{6} + 3480 p^{12} T^{7} + p^{16} T^{8} )^{2}
71 1−43758904T2+1149222526263836T4− 1 - 43758904 T^{2} + 1149222526263836 T^{4} - 38 ⁣⋯ ⁣0838\!\cdots\!08T6+ T^{6} + 12 ⁣⋯ ⁣8612\!\cdots\!86T8− T^{8} - 38 ⁣⋯ ⁣0838\!\cdots\!08p8T10+1149222526263836p16T12−43758904p24T14+p32T16 p^{8} T^{10} + 1149222526263836 p^{16} T^{12} - 43758904 p^{24} T^{14} + p^{32} T^{16}
73 (1−13824T+129639884T2−924716795904T3+5270348883746598T4−924716795904p4T5+129639884p8T6−13824p12T7+p16T8)2 ( 1 - 13824 T + 129639884 T^{2} - 924716795904 T^{3} + 5270348883746598 T^{4} - 924716795904 p^{4} T^{5} + 129639884 p^{8} T^{6} - 13824 p^{12} T^{7} + p^{16} T^{8} )^{2}
79 (1+512T+115971620T2+58605823488T3+6343534677753926T4+58605823488p4T5+115971620p8T6+512p12T7+p16T8)2 ( 1 + 512 T + 115971620 T^{2} + 58605823488 T^{3} + 6343534677753926 T^{4} + 58605823488 p^{4} T^{5} + 115971620 p^{8} T^{6} + 512 p^{12} T^{7} + p^{16} T^{8} )^{2}
83 1−190162744T2+15994881716551652T4− 1 - 190162744 T^{2} + 15994881716551652 T^{4} - 80 ⁣⋯ ⁣6080\!\cdots\!60T6+ T^{6} + 35 ⁣⋯ ⁣3835\!\cdots\!38T8− T^{8} - 80 ⁣⋯ ⁣6080\!\cdots\!60p8T10+15994881716551652p16T12−190162744p24T14+p32T16 p^{8} T^{10} + 15994881716551652 p^{16} T^{12} - 190162744 p^{24} T^{14} + p^{32} T^{16}
89 1−240564248T2+37258790807545948T4− 1 - 240564248 T^{2} + 37258790807545948 T^{4} - 36 ⁣⋯ ⁣7236\!\cdots\!72T6+ T^{6} + 27 ⁣⋯ ⁣8627\!\cdots\!86T8− T^{8} - 36 ⁣⋯ ⁣7236\!\cdots\!72p8T10+37258790807545948p16T12−240564248p24T14+p32T16 p^{8} T^{10} + 37258790807545948 p^{16} T^{12} - 240564248 p^{24} T^{14} + p^{32} T^{16}
97 (1+13704T+113687708T2+121903428792T3−333706776381114T4+121903428792p4T5+113687708p8T6+13704p12T7+p16T8)2 ( 1 + 13704 T + 113687708 T^{2} + 121903428792 T^{3} - 333706776381114 T^{4} + 121903428792 p^{4} T^{5} + 113687708 p^{8} T^{6} + 13704 p^{12} T^{7} + p^{16} T^{8} )^{2}
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   L(s)=∏p ∏j=116(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−6.94477191448057933739302706322, −6.94352217699111046347287990663, −6.66671921458207778148184959048, −6.38916709202995514804328157851, −6.24554211601366264253457498554, −5.93125375260446459406446752095, −5.38908138941423016707213879592, −5.22202016127420124349226771956, −5.15900607126687867869351658318, −5.15174021553366816964319172085, −4.68482167815059195726229074543, −4.66688584780610086941732400347, −4.54027947635743140789482835043, −4.20306223735757317565131996215, −3.68580370284497017265600537452, −3.55083575373499602034637554591, −3.48575976348390378866342428201, −2.79383347038542062982775082099, −2.77629435352663285337587270359, −2.68773701569136997878060250287, −2.23491538136301964201733568251, −1.43740858706091604531327737753, −1.19816482167351549059170056366, −0.71359067118230111325631782161, −0.50557307915793631667769541787, 0.50557307915793631667769541787, 0.71359067118230111325631782161, 1.19816482167351549059170056366, 1.43740858706091604531327737753, 2.23491538136301964201733568251, 2.68773701569136997878060250287, 2.77629435352663285337587270359, 2.79383347038542062982775082099, 3.48575976348390378866342428201, 3.55083575373499602034637554591, 3.68580370284497017265600537452, 4.20306223735757317565131996215, 4.54027947635743140789482835043, 4.66688584780610086941732400347, 4.68482167815059195726229074543, 5.15174021553366816964319172085, 5.15900607126687867869351658318, 5.22202016127420124349226771956, 5.38908138941423016707213879592, 5.93125375260446459406446752095, 6.24554211601366264253457498554, 6.38916709202995514804328157851, 6.66671921458207778148184959048, 6.94352217699111046347287990663, 6.94477191448057933739302706322

Graph of the ZZ-function along the critical line

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