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Properties

Label 16-1323e8-1.1-c3e8-0-1
Degree 1616
Conductor 9.386×10249.386\times 10^{24}
Sign 11
Analytic cond. 1.37850×10151.37850\times 10^{15}
Root an. cond. 8.835138.83513
Motivic weight 33
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  − 10·4-s + 336·13-s + 13·16-s − 288·19-s − 424·25-s + 120·31-s + 592·37-s − 1.87e3·43-s − 3.36e3·52-s + 2.40e3·61-s − 300·64-s + 1.82e3·73-s + 2.88e3·76-s + 2.36e3·79-s + 5.71e3·97-s + 4.24e3·100-s − 5.40e3·103-s − 1.65e3·109-s − 5.78e3·121-s − 1.20e3·124-s + 127-s + 131-s + 137-s + 139-s − 5.92e3·148-s + 149-s + 151-s + ⋯
L(s)  = 1  − 5/4·4-s + 7.16·13-s + 0.203·16-s − 3.47·19-s − 3.39·25-s + 0.695·31-s + 2.63·37-s − 6.63·43-s − 8.96·52-s + 5.03·61-s − 0.585·64-s + 2.92·73-s + 4.34·76-s + 3.37·79-s + 5.97·97-s + 4.23·100-s − 5.16·103-s − 1.45·109-s − 4.34·121-s − 0.869·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 3.28·148-s + 0.000549·149-s + 0.000538·151-s + ⋯

Functional equation

Λ(s)=((324⋅716)s/2 ΓC(s)8 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((324⋅716)s/2 ΓC(s+3/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 324⋅7163^{24} \cdot 7^{16}
Sign: 11
Analytic conductor: 1.37850×10151.37850\times 10^{15}
Root analytic conductor: 8.835138.83513
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 324⋅716, ( :[3/2]8), 1)(16,\ 3^{24} \cdot 7^{16} ,\ ( \ : [3/2]^{8} ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 2.7144856222.714485622
L(12)L(\frac12) ≈\approx 2.7144856222.714485622
L(52)L(\frac{5}{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 1
7 1 1
good2 1+5pT2+87T4+65p4T6+167p6T8+65p10T10+87p12T12+5p19T14+p24T16 1 + 5 p T^{2} + 87 T^{4} + 65 p^{4} T^{6} + 167 p^{6} T^{8} + 65 p^{10} T^{10} + 87 p^{12} T^{12} + 5 p^{19} T^{14} + p^{24} T^{16}
5 1+424T2+113862T4+21538016T6+3080570771T8+21538016p6T10+113862p12T12+424p18T14+p24T16 1 + 424 T^{2} + 113862 T^{4} + 21538016 T^{6} + 3080570771 T^{8} + 21538016 p^{6} T^{10} + 113862 p^{12} T^{12} + 424 p^{18} T^{14} + p^{24} T^{16}
11 1+5784T2+14142406T4+20701768608T6+25999720964691T8+20701768608p6T10+14142406p12T12+5784p18T14+p24T16 1 + 5784 T^{2} + 14142406 T^{4} + 20701768608 T^{6} + 25999720964691 T^{8} + 20701768608 p^{6} T^{10} + 14142406 p^{12} T^{12} + 5784 p^{18} T^{14} + p^{24} T^{16}
13 (1−168T+17116T2−1207896T3+64621222T4−1207896p3T5+17116p6T6−168p9T7+p12T8)2 ( 1 - 168 T + 17116 T^{2} - 1207896 T^{3} + 64621222 T^{4} - 1207896 p^{3} T^{5} + 17116 p^{6} T^{6} - 168 p^{9} T^{7} + p^{12} T^{8} )^{2}
17 1+4320T2+90997492T4+307024797600T6+3233455864654758T8+307024797600p6T10+90997492p12T12+4320p18T14+p24T16 1 + 4320 T^{2} + 90997492 T^{4} + 307024797600 T^{6} + 3233455864654758 T^{8} + 307024797600 p^{6} T^{10} + 90997492 p^{12} T^{12} + 4320 p^{18} T^{14} + p^{24} T^{16}
19 (1+144T+14886T2+829440T3+56352755T4+829440p3T5+14886p6T6+144p9T7+p12T8)2 ( 1 + 144 T + 14886 T^{2} + 829440 T^{3} + 56352755 T^{4} + 829440 p^{3} T^{5} + 14886 p^{6} T^{6} + 144 p^{9} T^{7} + p^{12} T^{8} )^{2}
23 1+37768T2+966746070T4+17415830892128T6+242639055368418659T8+17415830892128p6T10+966746070p12T12+37768p18T14+p24T16 1 + 37768 T^{2} + 966746070 T^{4} + 17415830892128 T^{6} + 242639055368418659 T^{8} + 17415830892128 p^{6} T^{10} + 966746070 p^{12} T^{12} + 37768 p^{18} T^{14} + p^{24} T^{16}
29 1+75064T2+4197383004T4+150911401826504T6+4338189011053329830T8+150911401826504p6T10+4197383004p12T12+75064p18T14+p24T16 1 + 75064 T^{2} + 4197383004 T^{4} + 150911401826504 T^{6} + 4338189011053329830 T^{8} + 150911401826504 p^{6} T^{10} + 4197383004 p^{12} T^{12} + 75064 p^{18} T^{14} + p^{24} T^{16}
31 (1−60T+72250T2−261480pT3+80919757pT4−261480p4T5+72250p6T6−60p9T7+p12T8)2 ( 1 - 60 T + 72250 T^{2} - 261480 p T^{3} + 80919757 p T^{4} - 261480 p^{4} T^{5} + 72250 p^{6} T^{6} - 60 p^{9} T^{7} + p^{12} T^{8} )^{2}
37 (1−8pT+44614T2+138976pT3−2636743133T4+138976p4T5+44614p6T6−8p10T7+p12T8)2 ( 1 - 8 p T + 44614 T^{2} + 138976 p T^{3} - 2636743133 T^{4} + 138976 p^{4} T^{5} + 44614 p^{6} T^{6} - 8 p^{10} T^{7} + p^{12} T^{8} )^{2}
41 1+388344T2+69574817686T4+7799366016841248T6+ 1 + 388344 T^{2} + 69574817686 T^{4} + 7799366016841248 T^{6} + 62 ⁣⋯ ⁣3162\!\cdots\!31T8+7799366016841248p6T10+69574817686p12T12+388344p18T14+p24T16 T^{8} + 7799366016841248 p^{6} T^{10} + 69574817686 p^{12} T^{12} + 388344 p^{18} T^{14} + p^{24} T^{16}
43 (1+936T+590184T2+242030232T3+79565108834T4+242030232p3T5+590184p6T6+936p9T7+p12T8)2 ( 1 + 936 T + 590184 T^{2} + 242030232 T^{3} + 79565108834 T^{4} + 242030232 p^{3} T^{5} + 590184 p^{6} T^{6} + 936 p^{9} T^{7} + p^{12} T^{8} )^{2}
47 1+539192T2+147167990020T4+25820308461522152T6+ 1 + 539192 T^{2} + 147167990020 T^{4} + 25820308461522152 T^{6} + 31 ⁣⋯ ⁣1431\!\cdots\!14T8+25820308461522152p6T10+147167990020p12T12+539192p18T14+p24T16 T^{8} + 25820308461522152 p^{6} T^{10} + 147167990020 p^{12} T^{12} + 539192 p^{18} T^{14} + p^{24} T^{16}
53 1+564888T2+122098902844T4+11434901409644904T6+ 1 + 564888 T^{2} + 122098902844 T^{4} + 11434901409644904 T^{6} + 73 ⁣⋯ ⁣6673\!\cdots\!66T8+11434901409644904p6T10+122098902844p12T12+564888p18T14+p24T16 T^{8} + 11434901409644904 p^{6} T^{10} + 122098902844 p^{12} T^{12} + 564888 p^{18} T^{14} + p^{24} T^{16}
59 1+502456T2+216666468900T4+63882710147039144T6+ 1 + 502456 T^{2} + 216666468900 T^{4} + 63882710147039144 T^{6} + 14 ⁣⋯ ⁣1414\!\cdots\!14T8+63882710147039144p6T10+216666468900p12T12+502456p18T14+p24T16 T^{8} + 63882710147039144 p^{6} T^{10} + 216666468900 p^{12} T^{12} + 502456 p^{18} T^{14} + p^{24} T^{16}
61 (1−1200T+1249988T2−843411120T3+464858253798T4−843411120p3T5+1249988p6T6−1200p9T7+p12T8)2 ( 1 - 1200 T + 1249988 T^{2} - 843411120 T^{3} + 464858253798 T^{4} - 843411120 p^{3} T^{5} + 1249988 p^{6} T^{6} - 1200 p^{9} T^{7} + p^{12} T^{8} )^{2}
67 (1+1023396T2−24675840T3+433884187622T4−24675840p3T5+1023396p6T6+p12T8)2 ( 1 + 1023396 T^{2} - 24675840 T^{3} + 433884187622 T^{4} - 24675840 p^{3} T^{5} + 1023396 p^{6} T^{6} + p^{12} T^{8} )^{2}
71 1+1910344T2+1810571288406T4+1096791110591321888T6+ 1 + 1910344 T^{2} + 1810571288406 T^{4} + 1096791110591321888 T^{6} + 46 ⁣⋯ ⁣9146\!\cdots\!91T8+1096791110591321888p6T10+1810571288406p12T12+1910344p18T14+p24T16 T^{8} + 1096791110591321888 p^{6} T^{10} + 1810571288406 p^{12} T^{12} + 1910344 p^{18} T^{14} + p^{24} T^{16}
73 (1−912T+1041832T2−557617680T3+474743673106T4−557617680p3T5+1041832p6T6−912p9T7+p12T8)2 ( 1 - 912 T + 1041832 T^{2} - 557617680 T^{3} + 474743673106 T^{4} - 557617680 p^{3} T^{5} + 1041832 p^{6} T^{6} - 912 p^{9} T^{7} + p^{12} T^{8} )^{2}
79 (1−1184T+2396136T2−1798718560T3+1877500132370T4−1798718560p3T5+2396136p6T6−1184p9T7+p12T8)2 ( 1 - 1184 T + 2396136 T^{2} - 1798718560 T^{3} + 1877500132370 T^{4} - 1798718560 p^{3} T^{5} + 2396136 p^{6} T^{6} - 1184 p^{9} T^{7} + p^{12} T^{8} )^{2}
83 1+3779800T2+6638776535772T4+7032576780729679400T6+ 1 + 3779800 T^{2} + 6638776535772 T^{4} + 7032576780729679400 T^{6} + 49 ⁣⋯ ⁣9849\!\cdots\!98T8+7032576780729679400p6T10+6638776535772p12T12+3779800p18T14+p24T16 T^{8} + 7032576780729679400 p^{6} T^{10} + 6638776535772 p^{12} T^{12} + 3779800 p^{18} T^{14} + p^{24} T^{16}
89 1+1909176T2+3175905142966T4+3100144053826523232T6+ 1 + 1909176 T^{2} + 3175905142966 T^{4} + 3100144053826523232 T^{6} + 26 ⁣⋯ ⁣1126\!\cdots\!11T8+3100144053826523232p6T10+3175905142966p12T12+1909176p18T14+p24T16 T^{8} + 3100144053826523232 p^{6} T^{10} + 3175905142966 p^{12} T^{12} + 1909176 p^{18} T^{14} + p^{24} T^{16}
97 (1−2856T+5804884T2−7716365688T3+8573207370022T4−7716365688p3T5+5804884p6T6−2856p9T7+p12T8)2 ( 1 - 2856 T + 5804884 T^{2} - 7716365688 T^{3} + 8573207370022 T^{4} - 7716365688 p^{3} T^{5} + 5804884 p^{6} T^{6} - 2856 p^{9} T^{7} + p^{12} 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

−3.85801123059411121913492777103, −3.67180858392766680597733582261, −3.49611404819188545812844493654, −3.48045965269665244402197762674, −3.37312515023666551480724375554, −3.04558551265308480324268362756, −2.86773863087654842400470117102, −2.80673665770956677217644193191, −2.70888079683286809722394627717, −2.39198215531027965805993533807, −2.23286718895847929345462738290, −2.13612829115830924689435886502, −2.11059585158939446179143385588, −1.69223889413001091209951861786, −1.64727491552508987929288088891, −1.56985479462041078611502291291, −1.55240959726617889010219230939, −1.38860969684012377389721046098, −1.19437958284839200176063101959, −0.907769866796201208928267673463, −0.63273476804702374096450144198, −0.58326202738954682314329749742, −0.49965696433510390071892428664, −0.47755916501800836198591919688, −0.07471588277330571986434297865, 0.07471588277330571986434297865, 0.47755916501800836198591919688, 0.49965696433510390071892428664, 0.58326202738954682314329749742, 0.63273476804702374096450144198, 0.907769866796201208928267673463, 1.19437958284839200176063101959, 1.38860969684012377389721046098, 1.55240959726617889010219230939, 1.56985479462041078611502291291, 1.64727491552508987929288088891, 1.69223889413001091209951861786, 2.11059585158939446179143385588, 2.13612829115830924689435886502, 2.23286718895847929345462738290, 2.39198215531027965805993533807, 2.70888079683286809722394627717, 2.80673665770956677217644193191, 2.86773863087654842400470117102, 3.04558551265308480324268362756, 3.37312515023666551480724375554, 3.48045965269665244402197762674, 3.49611404819188545812844493654, 3.67180858392766680597733582261, 3.85801123059411121913492777103

Graph of the ZZ-function along the critical line

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