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Properties

Label 16-810e8-1.1-c2e8-0-5
Degree 1616
Conductor 1.853×10231.853\times 10^{23}
Sign 11
Analytic cond. 5.63067×10105.63067\times 10^{10}
Root an. cond. 4.697964.69796
Motivic weight 22
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  − 4·4-s − 12·5-s + 4·16-s − 24·17-s − 24·19-s + 48·20-s − 60·23-s + 78·25-s + 68·31-s + 240·47-s − 8·49-s + 408·53-s + 196·61-s + 16·64-s + 96·68-s + 96·76-s − 180·79-s − 48·80-s − 108·83-s + 288·85-s + 240·92-s + 288·95-s − 312·100-s − 288·107-s − 152·109-s − 48·113-s + 720·115-s + ⋯
L(s)  = 1  − 4-s − 2.39·5-s + 1/4·16-s − 1.41·17-s − 1.26·19-s + 12/5·20-s − 2.60·23-s + 3.11·25-s + 2.19·31-s + 5.10·47-s − 0.163·49-s + 7.69·53-s + 3.21·61-s + 1/4·64-s + 1.41·68-s + 1.26·76-s − 2.27·79-s − 3/5·80-s − 1.30·83-s + 3.38·85-s + 2.60·92-s + 3.03·95-s − 3.11·100-s − 2.69·107-s − 1.39·109-s − 0.424·113-s + 6.26·115-s + ⋯

Functional equation

Λ(s)=((28⋅332⋅58)s/2 ΓC(s)8 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((28⋅332⋅58)s/2 ΓC(s+1)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 28⋅332⋅582^{8} \cdot 3^{32} \cdot 5^{8}
Sign: 11
Analytic conductor: 5.63067×10105.63067\times 10^{10}
Root analytic conductor: 4.697964.69796
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 28⋅332⋅58, ( :[1]8), 1)(16,\ 2^{8} \cdot 3^{32} \cdot 5^{8} ,\ ( \ : [1]^{8} ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.93023339480.9302333948
L(12)L(\frac12) ≈\approx 0.93023339480.9302333948
L(2)L(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)
bad2 (1+pT2+p2T4)2 ( 1 + p T^{2} + p^{2} T^{4} )^{2}
3 1 1
5 1+12T+66T2+336T3+1859T4+336p2T5+66p4T6+12p6T7+p8T8 1 + 12 T + 66 T^{2} + 336 T^{3} + 1859 T^{4} + 336 p^{2} T^{5} + 66 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8}
good7 1+8T2+3958T4−69568T6+9244771T8−69568p4T10+3958p8T12+8p12T14+p16T16 1 + 8 T^{2} + 3958 T^{4} - 69568 T^{6} + 9244771 T^{8} - 69568 p^{4} T^{10} + 3958 p^{8} T^{12} + 8 p^{12} T^{14} + p^{16} T^{16}
11 1+96T2−170p2T4+48384T6+537915459T8+48384p4T10−170p10T12+96p12T14+p16T16 1 + 96 T^{2} - 170 p^{2} T^{4} + 48384 T^{6} + 537915459 T^{8} + 48384 p^{4} T^{10} - 170 p^{10} T^{12} + 96 p^{12} T^{14} + p^{16} T^{16}
13 1+352T2+52006T4+5201152T6+814769539T8+5201152p4T10+52006p8T12+352p12T14+p16T16 1 + 352 T^{2} + 52006 T^{4} + 5201152 T^{6} + 814769539 T^{8} + 5201152 p^{4} T^{10} + 52006 p^{8} T^{12} + 352 p^{12} T^{14} + p^{16} T^{16}
17 (1+6T+489T2+6p2T3+p4T4)4 ( 1 + 6 T + 489 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{4}
19 (1+6T+713T2+6p2T3+p4T4)4 ( 1 + 6 T + 713 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{4}
23 (1+30T+9T2−5010T3+37940T4−5010p2T5+9p4T6+30p6T7+p8T8)2 ( 1 + 30 T + 9 T^{2} - 5010 T^{3} + 37940 T^{4} - 5010 p^{2} T^{5} + 9 p^{4} T^{6} + 30 p^{6} T^{7} + p^{8} T^{8} )^{2}
29 1+1096T2−41258T4−188608448T6+193897974019T8−188608448p4T10−41258p8T12+1096p12T14+p16T16 1 + 1096 T^{2} - 41258 T^{4} - 188608448 T^{6} + 193897974019 T^{8} - 188608448 p^{4} T^{10} - 41258 p^{8} T^{12} + 1096 p^{12} T^{14} + p^{16} T^{16}
31 (1−34T−605T2+5474T3+1066684T4+5474p2T5−605p4T6−34p6T7+p8T8)2 ( 1 - 34 T - 605 T^{2} + 5474 T^{3} + 1066684 T^{4} + 5474 p^{2} T^{5} - 605 p^{4} T^{6} - 34 p^{6} T^{7} + p^{8} T^{8} )^{2}
37 (1−68T2−1268634T4−68p4T6+p8T8)2 ( 1 - 68 T^{2} - 1268634 T^{4} - 68 p^{4} T^{6} + p^{8} T^{8} )^{2}
41 1+5352T2+16236406T4+36157983552T6+65020254579939T8+36157983552p4T10+16236406p8T12+5352p12T14+p16T16 1 + 5352 T^{2} + 16236406 T^{4} + 36157983552 T^{6} + 65020254579939 T^{8} + 36157983552 p^{4} T^{10} + 16236406 p^{8} T^{12} + 5352 p^{12} T^{14} + p^{16} T^{16}
43 1+5672T2+17968534T4+41776821056T6+80050185474115T8+41776821056p4T10+17968534p8T12+5672p12T14+p16T16 1 + 5672 T^{2} + 17968534 T^{4} + 41776821056 T^{6} + 80050185474115 T^{8} + 41776821056 p^{4} T^{10} + 17968534 p^{8} T^{12} + 5672 p^{12} T^{14} + p^{16} T^{16}
47 (1−120T+6774T2−384960T3+21466595T4−384960p2T5+6774p4T6−120p6T7+p8T8)2 ( 1 - 120 T + 6774 T^{2} - 384960 T^{3} + 21466595 T^{4} - 384960 p^{2} T^{5} + 6774 p^{4} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} )^{2}
53 (1−102T+8057T2−102p2T3+p4T4)4 ( 1 - 102 T + 8057 T^{2} - 102 p^{2} T^{3} + p^{4} T^{4} )^{4}
59 1+10848T2+64062694T4+318732551424T6+1301928798359235T8+318732551424p4T10+64062694p8T12+10848p12T14+p16T16 1 + 10848 T^{2} + 64062694 T^{4} + 318732551424 T^{6} + 1301928798359235 T^{8} + 318732551424 p^{4} T^{10} + 64062694 p^{8} T^{12} + 10848 p^{12} T^{14} + p^{16} T^{16}
61 (1−98T+913T2−122402T3+25951156T4−122402p2T5+913p4T6−98p6T7+p8T8)2 ( 1 - 98 T + 913 T^{2} - 122402 T^{3} + 25951156 T^{4} - 122402 p^{2} T^{5} + 913 p^{4} T^{6} - 98 p^{6} T^{7} + p^{8} T^{8} )^{2}
67 1−6508T2+10043914T4+52012534736T6−215603753039885T8+52012534736p4T10+10043914p8T12−6508p12T14+p16T16 1 - 6508 T^{2} + 10043914 T^{4} + 52012534736 T^{6} - 215603753039885 T^{8} + 52012534736 p^{4} T^{10} + 10043914 p^{8} T^{12} - 6508 p^{12} T^{14} + p^{16} T^{16}
71 (1−4288T2+45932730T4−4288p4T6+p8T8)2 ( 1 - 4288 T^{2} + 45932730 T^{4} - 4288 p^{4} T^{6} + p^{8} T^{8} )^{2}
73 (1−13280T2+84777594T4−13280p4T6+p8T8)2 ( 1 - 13280 T^{2} + 84777594 T^{4} - 13280 p^{4} T^{6} + p^{8} T^{8} )^{2}
79 (1+90T+1531T2−532170T3−46350420T4−532170p2T5+1531p4T6+90p6T7+p8T8)2 ( 1 + 90 T + 1531 T^{2} - 532170 T^{3} - 46350420 T^{4} - 532170 p^{2} T^{5} + 1531 p^{4} T^{6} + 90 p^{6} T^{7} + p^{8} T^{8} )^{2}
83 (1+54T−4863T2−323946T3−7033804T4−323946p2T5−4863p4T6+54p6T7+p8T8)2 ( 1 + 54 T - 4863 T^{2} - 323946 T^{3} - 7033804 T^{4} - 323946 p^{2} T^{5} - 4863 p^{4} T^{6} + 54 p^{6} T^{7} + p^{8} T^{8} )^{2}
89 (1−8136T2+91108874T4−8136p4T6+p8T8)2 ( 1 - 8136 T^{2} + 91108874 T^{4} - 8136 p^{4} T^{6} + p^{8} T^{8} )^{2}
97 1+19172T2+99652426T4+1741864314512T6+31399709763290899T8+1741864314512p4T10+99652426p8T12+19172p12T14+p16T16 1 + 19172 T^{2} + 99652426 T^{4} + 1741864314512 T^{6} + 31399709763290899 T^{8} + 1741864314512 p^{4} T^{10} + 99652426 p^{8} T^{12} + 19172 p^{12} T^{14} + p^{16} T^{16}
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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

−4.10804013910753723116328626251, −4.04741587646894709786066585647, −4.02335260674913528056436148356, −4.00927154651713340206162425555, −3.96991360697020507313288918671, −3.77756894658837106240983324333, −3.62847622748312572550362252916, −3.03548737831584495105297620840, −2.95164571100874786191989904165, −2.94399051906611577100944918549, −2.81393725788715210346190998928, −2.76688406007992074523015923868, −2.55523594474710831199619177002, −2.27139735458653082975685804872, −2.09138761141746423621700220638, −2.03140740518457969547061455779, −1.99201252381411382649445591788, −1.81074646093997913882376712388, −1.12439497969789112115104229166, −1.07220884815496910694421035658, −0.932099603229460782758466169489, −0.78493788701566893907115934662, −0.59573891358505706121519967929, −0.28349056199335134225424521840, −0.17087934155385081822813630987, 0.17087934155385081822813630987, 0.28349056199335134225424521840, 0.59573891358505706121519967929, 0.78493788701566893907115934662, 0.932099603229460782758466169489, 1.07220884815496910694421035658, 1.12439497969789112115104229166, 1.81074646093997913882376712388, 1.99201252381411382649445591788, 2.03140740518457969547061455779, 2.09138761141746423621700220638, 2.27139735458653082975685804872, 2.55523594474710831199619177002, 2.76688406007992074523015923868, 2.81393725788715210346190998928, 2.94399051906611577100944918549, 2.95164571100874786191989904165, 3.03548737831584495105297620840, 3.62847622748312572550362252916, 3.77756894658837106240983324333, 3.96991360697020507313288918671, 4.00927154651713340206162425555, 4.02335260674913528056436148356, 4.04741587646894709786066585647, 4.10804013910753723116328626251

Graph of the ZZ-function along the critical line

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