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Properties

Label 16-7400e8-1.1-c1e8-0-3
Degree 1616
Conductor 8.992×10308.992\times 10^{30}
Sign 11
Analytic cond. 1.48617×10141.48617\times 10^{14}
Root an. cond. 7.686957.68695
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 88

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·7-s − 11·9-s − 9·13-s − 4·17-s + 4·19-s + 4·21-s − 2·23-s + 10·27-s − 5·29-s + 11·31-s + 8·37-s + 9·39-s − 4·43-s − 14·47-s − 25·49-s + 4·51-s − 11·53-s − 4·57-s − 15·59-s − 13·61-s + 44·63-s − 19·67-s + 2·69-s + 40·71-s − 31·73-s + 2·79-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s − 3.66·9-s − 2.49·13-s − 0.970·17-s + 0.917·19-s + 0.872·21-s − 0.417·23-s + 1.92·27-s − 0.928·29-s + 1.97·31-s + 1.31·37-s + 1.44·39-s − 0.609·43-s − 2.04·47-s − 3.57·49-s + 0.560·51-s − 1.51·53-s − 0.529·57-s − 1.95·59-s − 1.66·61-s + 5.54·63-s − 2.32·67-s + 0.240·69-s + 4.74·71-s − 3.62·73-s + 0.225·79-s + ⋯

Functional equation

Λ(s)=((224⋅516⋅378)s/2 ΓC(s)8 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{16} \cdot 37^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((224⋅516⋅378)s/2 ΓC(s+1/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{16} \cdot 37^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 224⋅516⋅3782^{24} \cdot 5^{16} \cdot 37^{8}
Sign: 11
Analytic conductor: 1.48617×10141.48617\times 10^{14}
Root analytic conductor: 7.686957.68695
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 88
Selberg data: (16, 224⋅516⋅378, ( :[1/2]8), 1)(16,\ 2^{24} \cdot 5^{16} \cdot 37^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad2 1 1
5 1 1
37 (1−T)8 ( 1 - T )^{8}
good3 1+T+4pT2+13T3+25pT4+83T5+4p4T6+340T7+362pT8+340pT9+4p6T10+83p3T11+25p5T12+13p5T13+4p7T14+p7T15+p8T16 1 + T + 4 p T^{2} + 13 T^{3} + 25 p T^{4} + 83 T^{5} + 4 p^{4} T^{6} + 340 T^{7} + 362 p T^{8} + 340 p T^{9} + 4 p^{6} T^{10} + 83 p^{3} T^{11} + 25 p^{5} T^{12} + 13 p^{5} T^{13} + 4 p^{7} T^{14} + p^{7} T^{15} + p^{8} T^{16}
7 1+4T+41T2+134T3+760T4+2141T5+8811T6+3103pT7+72190T8+3103p2T9+8811p2T10+2141p3T11+760p4T12+134p5T13+41p6T14+4p7T15+p8T16 1 + 4 T + 41 T^{2} + 134 T^{3} + 760 T^{4} + 2141 T^{5} + 8811 T^{6} + 3103 p T^{7} + 72190 T^{8} + 3103 p^{2} T^{9} + 8811 p^{2} T^{10} + 2141 p^{3} T^{11} + 760 p^{4} T^{12} + 134 p^{5} T^{13} + 41 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16}
11 1+40T2+21T3+756T4+1183T5+9179T6+24891T7+95896T8+24891pT9+9179p2T10+1183p3T11+756p4T12+21p5T13+40p6T14+p8T16 1 + 40 T^{2} + 21 T^{3} + 756 T^{4} + 1183 T^{5} + 9179 T^{6} + 24891 T^{7} + 95896 T^{8} + 24891 p T^{9} + 9179 p^{2} T^{10} + 1183 p^{3} T^{11} + 756 p^{4} T^{12} + 21 p^{5} T^{13} + 40 p^{6} T^{14} + p^{8} T^{16}
13 1+9T+99T2+634T3+4216T4+20993T5+104419T6+419854T7+1666702T8+419854pT9+104419p2T10+20993p3T11+4216p4T12+634p5T13+99p6T14+9p7T15+p8T16 1 + 9 T + 99 T^{2} + 634 T^{3} + 4216 T^{4} + 20993 T^{5} + 104419 T^{6} + 419854 T^{7} + 1666702 T^{8} + 419854 p T^{9} + 104419 p^{2} T^{10} + 20993 p^{3} T^{11} + 4216 p^{4} T^{12} + 634 p^{5} T^{13} + 99 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16}
17 1+4T+71T2+145T3+2168T4+1557T5+2687pT6−217T7+823058T8−217pT9+2687p3T10+1557p3T11+2168p4T12+145p5T13+71p6T14+4p7T15+p8T16 1 + 4 T + 71 T^{2} + 145 T^{3} + 2168 T^{4} + 1557 T^{5} + 2687 p T^{6} - 217 T^{7} + 823058 T^{8} - 217 p T^{9} + 2687 p^{3} T^{10} + 1557 p^{3} T^{11} + 2168 p^{4} T^{12} + 145 p^{5} T^{13} + 71 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16}
19 1−4T+89T2−356T3+3589T4−15109T5+92127T6−407004T7+1868187T8−407004pT9+92127p2T10−15109p3T11+3589p4T12−356p5T13+89p6T14−4p7T15+p8T16 1 - 4 T + 89 T^{2} - 356 T^{3} + 3589 T^{4} - 15109 T^{5} + 92127 T^{6} - 407004 T^{7} + 1868187 T^{8} - 407004 p T^{9} + 92127 p^{2} T^{10} - 15109 p^{3} T^{11} + 3589 p^{4} T^{12} - 356 p^{5} T^{13} + 89 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16}
23 1+2T+94T2+301T3+4971T4+16391T5+182755T6+565688T7+4833338T8+565688pT9+182755p2T10+16391p3T11+4971p4T12+301p5T13+94p6T14+2p7T15+p8T16 1 + 2 T + 94 T^{2} + 301 T^{3} + 4971 T^{4} + 16391 T^{5} + 182755 T^{6} + 565688 T^{7} + 4833338 T^{8} + 565688 p T^{9} + 182755 p^{2} T^{10} + 16391 p^{3} T^{11} + 4971 p^{4} T^{12} + 301 p^{5} T^{13} + 94 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16}
29 1+5T+187T2+800T3+16086T4+58905T5+839763T6+2610422T7+29357790T8+2610422pT9+839763p2T10+58905p3T11+16086p4T12+800p5T13+187p6T14+5p7T15+p8T16 1 + 5 T + 187 T^{2} + 800 T^{3} + 16086 T^{4} + 58905 T^{5} + 839763 T^{6} + 2610422 T^{7} + 29357790 T^{8} + 2610422 p T^{9} + 839763 p^{2} T^{10} + 58905 p^{3} T^{11} + 16086 p^{4} T^{12} + 800 p^{5} T^{13} + 187 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16}
31 1−11T+231T2−1905T3+23233T4−153870T5+1371031T6−7421606T7+52211608T8−7421606pT9+1371031p2T10−153870p3T11+23233p4T12−1905p5T13+231p6T14−11p7T15+p8T16 1 - 11 T + 231 T^{2} - 1905 T^{3} + 23233 T^{4} - 153870 T^{5} + 1371031 T^{6} - 7421606 T^{7} + 52211608 T^{8} - 7421606 p T^{9} + 1371031 p^{2} T^{10} - 153870 p^{3} T^{11} + 23233 p^{4} T^{12} - 1905 p^{5} T^{13} + 231 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16}
41 1+224T2−141T3+24629T4−22712T5+1723583T6−1673307T7+83770693T8−1673307pT9+1723583p2T10−22712p3T11+24629p4T12−141p5T13+224p6T14+p8T16 1 + 224 T^{2} - 141 T^{3} + 24629 T^{4} - 22712 T^{5} + 1723583 T^{6} - 1673307 T^{7} + 83770693 T^{8} - 1673307 p T^{9} + 1723583 p^{2} T^{10} - 22712 p^{3} T^{11} + 24629 p^{4} T^{12} - 141 p^{5} T^{13} + 224 p^{6} T^{14} + p^{8} T^{16}
43 1+4T+76T2+193T3+5489T4+24195T5+355439T6+947630T7+14585842T8+947630pT9+355439p2T10+24195p3T11+5489p4T12+193p5T13+76p6T14+4p7T15+p8T16 1 + 4 T + 76 T^{2} + 193 T^{3} + 5489 T^{4} + 24195 T^{5} + 355439 T^{6} + 947630 T^{7} + 14585842 T^{8} + 947630 p T^{9} + 355439 p^{2} T^{10} + 24195 p^{3} T^{11} + 5489 p^{4} T^{12} + 193 p^{5} T^{13} + 76 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16}
47 1+14T+203T2+2202T3+22790T4+190523T5+1638719T6+11878747T7+86001310T8+11878747pT9+1638719p2T10+190523p3T11+22790p4T12+2202p5T13+203p6T14+14p7T15+p8T16 1 + 14 T + 203 T^{2} + 2202 T^{3} + 22790 T^{4} + 190523 T^{5} + 1638719 T^{6} + 11878747 T^{7} + 86001310 T^{8} + 11878747 p T^{9} + 1638719 p^{2} T^{10} + 190523 p^{3} T^{11} + 22790 p^{4} T^{12} + 2202 p^{5} T^{13} + 203 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16}
53 1+11T+229T2+1735T3+22744T4+123313T5+1409954T6+6077569T7+74018624T8+6077569pT9+1409954p2T10+123313p3T11+22744p4T12+1735p5T13+229p6T14+11p7T15+p8T16 1 + 11 T + 229 T^{2} + 1735 T^{3} + 22744 T^{4} + 123313 T^{5} + 1409954 T^{6} + 6077569 T^{7} + 74018624 T^{8} + 6077569 p T^{9} + 1409954 p^{2} T^{10} + 123313 p^{3} T^{11} + 22744 p^{4} T^{12} + 1735 p^{5} T^{13} + 229 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16}
59 1+15T+407T2+4664T3+74618T4+688341T5+8132690T6+61792736T7+583695956T8+61792736pT9+8132690p2T10+688341p3T11+74618p4T12+4664p5T13+407p6T14+15p7T15+p8T16 1 + 15 T + 407 T^{2} + 4664 T^{3} + 74618 T^{4} + 688341 T^{5} + 8132690 T^{6} + 61792736 T^{7} + 583695956 T^{8} + 61792736 p T^{9} + 8132690 p^{2} T^{10} + 688341 p^{3} T^{11} + 74618 p^{4} T^{12} + 4664 p^{5} T^{13} + 407 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16}
61 1+13T+163T2+1165T3+16487T4+142792T5+1563853T6+9653588T7+89810140T8+9653588pT9+1563853p2T10+142792p3T11+16487p4T12+1165p5T13+163p6T14+13p7T15+p8T16 1 + 13 T + 163 T^{2} + 1165 T^{3} + 16487 T^{4} + 142792 T^{5} + 1563853 T^{6} + 9653588 T^{7} + 89810140 T^{8} + 9653588 p T^{9} + 1563853 p^{2} T^{10} + 142792 p^{3} T^{11} + 16487 p^{4} T^{12} + 1165 p^{5} T^{13} + 163 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16}
67 1+19T+559T2+7561T3+127663T4+1338243T5+16489908T6+139349862T7+1359453328T8+139349862pT9+16489908p2T10+1338243p3T11+127663p4T12+7561p5T13+559p6T14+19p7T15+p8T16 1 + 19 T + 559 T^{2} + 7561 T^{3} + 127663 T^{4} + 1338243 T^{5} + 16489908 T^{6} + 139349862 T^{7} + 1359453328 T^{8} + 139349862 p T^{9} + 16489908 p^{2} T^{10} + 1338243 p^{3} T^{11} + 127663 p^{4} T^{12} + 7561 p^{5} T^{13} + 559 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16}
71 1−40T+899T2−14968T3+209296T4−2532989T5+27167623T6−262647011T7+2318120662T8−262647011pT9+27167623p2T10−2532989p3T11+209296p4T12−14968p5T13+899p6T14−40p7T15+p8T16 1 - 40 T + 899 T^{2} - 14968 T^{3} + 209296 T^{4} - 2532989 T^{5} + 27167623 T^{6} - 262647011 T^{7} + 2318120662 T^{8} - 262647011 p T^{9} + 27167623 p^{2} T^{10} - 2532989 p^{3} T^{11} + 209296 p^{4} T^{12} - 14968 p^{5} T^{13} + 899 p^{6} T^{14} - 40 p^{7} T^{15} + p^{8} T^{16}
73 1+31T+859T2+15266T3+247690T4+3153920T5+37367805T6+369631911T7+3429783898T8+369631911pT9+37367805p2T10+3153920p3T11+247690p4T12+15266p5T13+859p6T14+31p7T15+p8T16 1 + 31 T + 859 T^{2} + 15266 T^{3} + 247690 T^{4} + 3153920 T^{5} + 37367805 T^{6} + 369631911 T^{7} + 3429783898 T^{8} + 369631911 p T^{9} + 37367805 p^{2} T^{10} + 3153920 p^{3} T^{11} + 247690 p^{4} T^{12} + 15266 p^{5} T^{13} + 859 p^{6} T^{14} + 31 p^{7} T^{15} + p^{8} T^{16}
79 1−2T+241T2+317T3+25518T4+160045T5+1915546T6+21066986T7+144183384T8+21066986pT9+1915546p2T10+160045p3T11+25518p4T12+317p5T13+241p6T14−2p7T15+p8T16 1 - 2 T + 241 T^{2} + 317 T^{3} + 25518 T^{4} + 160045 T^{5} + 1915546 T^{6} + 21066986 T^{7} + 144183384 T^{8} + 21066986 p T^{9} + 1915546 p^{2} T^{10} + 160045 p^{3} T^{11} + 25518 p^{4} T^{12} + 317 p^{5} T^{13} + 241 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16}
83 1+4T+301T2+857T3+54134T4+95849T5+6417081T6+7294227T7+604402462T8+7294227pT9+6417081p2T10+95849p3T11+54134p4T12+857p5T13+301p6T14+4p7T15+p8T16 1 + 4 T + 301 T^{2} + 857 T^{3} + 54134 T^{4} + 95849 T^{5} + 6417081 T^{6} + 7294227 T^{7} + 604402462 T^{8} + 7294227 p T^{9} + 6417081 p^{2} T^{10} + 95849 p^{3} T^{11} + 54134 p^{4} T^{12} + 857 p^{5} T^{13} + 301 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16}
89 1+12T+459T2+4425T3+102144T4+851303T5+14968945T6+109132431T7+17613520pT8+109132431pT9+14968945p2T10+851303p3T11+102144p4T12+4425p5T13+459p6T14+12p7T15+p8T16 1 + 12 T + 459 T^{2} + 4425 T^{3} + 102144 T^{4} + 851303 T^{5} + 14968945 T^{6} + 109132431 T^{7} + 17613520 p T^{8} + 109132431 p T^{9} + 14968945 p^{2} T^{10} + 851303 p^{3} T^{11} + 102144 p^{4} T^{12} + 4425 p^{5} T^{13} + 459 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16}
97 1+11T+341T2+3139T3+37601T4+191096T5−201829T6−22517198T7−298694996T8−22517198pT9−201829p2T10+191096p3T11+37601p4T12+3139p5T13+341p6T14+11p7T15+p8T16 1 + 11 T + 341 T^{2} + 3139 T^{3} + 37601 T^{4} + 191096 T^{5} - 201829 T^{6} - 22517198 T^{7} - 298694996 T^{8} - 22517198 p T^{9} - 201829 p^{2} T^{10} + 191096 p^{3} T^{11} + 37601 p^{4} T^{12} + 3139 p^{5} T^{13} + 341 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} 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

−3.63781537093458223077848955815, −3.35358602094162403631116762944, −3.16866834861953392050049366688, −3.11060510655001701741078058305, −3.09263506739564643488899512771, −3.06323099618941609541453997654, −3.06085589887160388742016133900, −2.99852503550551270884118000171, −2.99734256247562265776340215907, −2.61434727497127814684422254156, −2.49599059664715106792597623423, −2.49253999766850301263740328006, −2.37758830003560983719967997036, −2.20968707240768147442259439429, −2.17833575734121189749515951834, −2.13389746703319454506044827658, −2.11093992332776658930164184034, −1.57990519623249631007725704567, −1.57425735451752144925734281963, −1.39545628431842754335440431609, −1.30669521302015643659372614173, −1.30388085362307466487649111931, −1.03752905638199006409131040907, −0.964386981791585366381639066511, −0.879741498050797199283884864291, 0, 0, 0, 0, 0, 0, 0, 0, 0.879741498050797199283884864291, 0.964386981791585366381639066511, 1.03752905638199006409131040907, 1.30388085362307466487649111931, 1.30669521302015643659372614173, 1.39545628431842754335440431609, 1.57425735451752144925734281963, 1.57990519623249631007725704567, 2.11093992332776658930164184034, 2.13389746703319454506044827658, 2.17833575734121189749515951834, 2.20968707240768147442259439429, 2.37758830003560983719967997036, 2.49253999766850301263740328006, 2.49599059664715106792597623423, 2.61434727497127814684422254156, 2.99734256247562265776340215907, 2.99852503550551270884118000171, 3.06085589887160388742016133900, 3.06323099618941609541453997654, 3.09263506739564643488899512771, 3.11060510655001701741078058305, 3.16866834861953392050049366688, 3.35358602094162403631116762944, 3.63781537093458223077848955815

Graph of the ZZ-function along the critical line

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