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Properties

Label 16-162e8-1.1-c6e8-0-1
Degree 1616
Conductor 4.744×10174.744\times 10^{17}
Sign 11
Analytic cond. 3.72185×10123.72185\times 10^{12}
Root an. cond. 6.104816.10481
Motivic weight 66
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  − 128·4-s + 964·7-s + 4.54e3·13-s + 1.02e4·16-s − 2.36e4·19-s + 4.63e4·25-s − 1.23e5·28-s + 7.70e4·31-s − 1.13e4·37-s − 2.26e5·43-s + 6.43e5·49-s − 5.81e5·52-s − 3.27e5·61-s − 6.55e5·64-s + 1.71e6·67-s − 2.18e6·73-s + 3.03e6·76-s − 1.32e6·79-s + 4.37e6·91-s − 2.20e6·97-s − 5.92e6·100-s − 4.49e4·103-s + 4.47e5·109-s + 9.87e6·112-s + 2.06e6·121-s − 9.86e6·124-s + 127-s + ⋯
L(s)  = 1  − 2·4-s + 2.81·7-s + 2.06·13-s + 5/2·16-s − 3.45·19-s + 2.96·25-s − 5.62·28-s + 2.58·31-s − 0.224·37-s − 2.85·43-s + 5.46·49-s − 4.13·52-s − 1.44·61-s − 5/2·64-s + 5.69·67-s − 5.62·73-s + 6.90·76-s − 2.69·79-s + 5.80·91-s − 2.41·97-s − 5.92·100-s − 0.0411·103-s + 0.345·109-s + 7.02·112-s + 1.16·121-s − 5.17·124-s − 9.70·133-s + ⋯

Functional equation

Λ(s)=((28⋅332)s/2 ΓC(s)8 L(s)=( Λ(7−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}
Λ(s)=((28⋅332)s/2 ΓC(s+3)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 28⋅3322^{8} \cdot 3^{32}
Sign: 11
Analytic conductor: 3.72185×10123.72185\times 10^{12}
Root analytic conductor: 6.104816.10481
Motivic weight: 66
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 28⋅332, ( :[3]8), 1)(16,\ 2^{8} \cdot 3^{32} ,\ ( \ : [3]^{8} ),\ 1 )

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 1.0939250341.093925034
L(12)L(\frac12) ≈\approx 1.0939250341.093925034
L(4)L(4) 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+p5T2)4 ( 1 + p^{5} T^{2} )^{4}
3 1 1
good5 1−46304T2+1650266686T4−59862115328p4T6+1769541248611p8T8−59862115328p16T10+1650266686p24T12−46304p36T14+p48T16 1 - 46304 T^{2} + 1650266686 T^{4} - 59862115328 p^{4} T^{6} + 1769541248611 p^{8} T^{8} - 59862115328 p^{16} T^{10} + 1650266686 p^{24} T^{12} - 46304 p^{36} T^{14} + p^{48} T^{16}
7 (1−482T+26938T2−17028398T3+18436500226T4−17028398p6T5+26938p12T6−482p18T7+p24T8)2 ( 1 - 482 T + 26938 T^{2} - 17028398 T^{3} + 18436500226 T^{4} - 17028398 p^{6} T^{5} + 26938 p^{12} T^{6} - 482 p^{18} T^{7} + p^{24} T^{8} )^{2}
11 1−2067740T2+3021523804504T4−2921892925855093364T6− 1 - 2067740 T^{2} + 3021523804504 T^{4} - 2921892925855093364 T^{6} - 30 ⁣⋯ ⁣2630\!\cdots\!26T8−2921892925855093364p12T10+3021523804504p24T12−2067740p36T14+p48T16 T^{8} - 2921892925855093364 p^{12} T^{10} + 3021523804504 p^{24} T^{12} - 2067740 p^{36} T^{14} + p^{48} T^{16}
13 (1−2270T+12646702T2−30153498176T3+78761957653363T4−30153498176p6T5+12646702p12T6−2270p18T7+p24T8)2 ( 1 - 2270 T + 12646702 T^{2} - 30153498176 T^{3} + 78761957653363 T^{4} - 30153498176 p^{6} T^{5} + 12646702 p^{12} T^{6} - 2270 p^{18} T^{7} + p^{24} T^{8} )^{2}
17 1−141183800T2+9651342187425574T4− 1 - 141183800 T^{2} + 9651342187425574 T^{4} - 41 ⁣⋯ ⁣6841\!\cdots\!68T6+ T^{6} + 11 ⁣⋯ ⁣3911\!\cdots\!39T8− T^{8} - 41 ⁣⋯ ⁣6841\!\cdots\!68p12T10+9651342187425574p24T12−141183800p36T14+p48T16 p^{12} T^{10} + 9651342187425574 p^{24} T^{12} - 141183800 p^{36} T^{14} + p^{48} T^{16}
19 (1+11842T+122570434T2+382269231070T3+2961485052850642T4+382269231070p6T5+122570434p12T6+11842p18T7+p24T8)2 ( 1 + 11842 T + 122570434 T^{2} + 382269231070 T^{3} + 2961485052850642 T^{4} + 382269231070 p^{6} T^{5} + 122570434 p^{12} T^{6} + 11842 p^{18} T^{7} + p^{24} T^{8} )^{2}
23 1−602193140T2+208847216279397736T4− 1 - 602193140 T^{2} + 208847216279397736 T^{4} - 21 ⁣⋯ ⁣2021\!\cdots\!20pT6+ p T^{6} + 84 ⁣⋯ ⁣0684\!\cdots\!06T8− T^{8} - 21 ⁣⋯ ⁣2021\!\cdots\!20p13T10+208847216279397736p24T12−602193140p36T14+p48T16 p^{13} T^{10} + 208847216279397736 p^{24} T^{12} - 602193140 p^{36} T^{14} + p^{48} T^{16}
29 1−3072361832T2+4718062571292062902T4− 1 - 3072361832 T^{2} + 4718062571292062902 T^{4} - 46 ⁣⋯ ⁣8046\!\cdots\!80T6+ T^{6} + 32 ⁣⋯ ⁣3932\!\cdots\!39T8− T^{8} - 46 ⁣⋯ ⁣8046\!\cdots\!80p12T10+4718062571292062902p24T12−3072361832p36T14+p48T16 p^{12} T^{10} + 4718062571292062902 p^{24} T^{12} - 3072361832 p^{36} T^{14} + p^{48} T^{16}
31 (1−38528T+2054564344T2−23224523393216T3+1280087508642426862T4−23224523393216p6T5+2054564344p12T6−38528p18T7+p24T8)2 ( 1 - 38528 T + 2054564344 T^{2} - 23224523393216 T^{3} + 1280087508642426862 T^{4} - 23224523393216 p^{6} T^{5} + 2054564344 p^{12} T^{6} - 38528 p^{18} T^{7} + p^{24} T^{8} )^{2}
37 (1+5674T+2614216798T2−33031702931096T3+7074013795851396475T4−33031702931096p6T5+2614216798p12T6+5674p18T7+p24T8)2 ( 1 + 5674 T + 2614216798 T^{2} - 33031702931096 T^{3} + 7074013795851396475 T^{4} - 33031702931096 p^{6} T^{5} + 2614216798 p^{12} T^{6} + 5674 p^{18} T^{7} + p^{24} T^{8} )^{2}
41 1−24361730072T2+ 1 - 24361730072 T^{2} + 25 ⁣⋯ ⁣8025\!\cdots\!80T4− T^{4} - 15 ⁣⋯ ⁣0015\!\cdots\!00T6+ T^{6} + 75 ⁣⋯ ⁣0675\!\cdots\!06T8− T^{8} - 15 ⁣⋯ ⁣0015\!\cdots\!00p12T10+ p^{12} T^{10} + 25 ⁣⋯ ⁣8025\!\cdots\!80p24T12−24361730072p36T14+p48T16 p^{24} T^{12} - 24361730072 p^{36} T^{14} + p^{48} T^{16}
43 (1+113302T+25372893394T2+975408578554p2T3+ ( 1 + 113302 T + 25372893394 T^{2} + 975408578554 p^{2} T^{3} + 23 ⁣⋯ ⁣5823\!\cdots\!58T4+975408578554p8T5+25372893394p12T6+113302p18T7+p24T8)2 T^{4} + 975408578554 p^{8} T^{5} + 25372893394 p^{12} T^{6} + 113302 p^{18} T^{7} + p^{24} T^{8} )^{2}
47 1−45886260248T2+ 1 - 45886260248 T^{2} + 10 ⁣⋯ ⁣4810\!\cdots\!48T4− T^{4} - 16 ⁣⋯ ⁣4416\!\cdots\!44T6+ T^{6} + 20 ⁣⋯ ⁣7420\!\cdots\!74T8− T^{8} - 16 ⁣⋯ ⁣4416\!\cdots\!44p12T10+ p^{12} T^{10} + 10 ⁣⋯ ⁣4810\!\cdots\!48p24T12−45886260248p36T14+p48T16 p^{24} T^{12} - 45886260248 p^{36} T^{14} + p^{48} T^{16}
53 1−97731632312T2+ 1 - 97731632312 T^{2} + 51 ⁣⋯ ⁣9251\!\cdots\!92T4− T^{4} - 18 ⁣⋯ ⁣0818\!\cdots\!08T6+ T^{6} + 47 ⁣⋯ ⁣1047\!\cdots\!10T8− T^{8} - 18 ⁣⋯ ⁣0818\!\cdots\!08p12T10+ p^{12} T^{10} + 51 ⁣⋯ ⁣9251\!\cdots\!92p24T12−97731632312p36T14+p48T16 p^{24} T^{12} - 97731632312 p^{36} T^{14} + p^{48} T^{16}
59 1−68754072824T2+ 1 - 68754072824 T^{2} + 28 ⁣⋯ ⁣8428\!\cdots\!84T4− T^{4} - 11 ⁣⋯ ⁣7211\!\cdots\!72T6+ T^{6} + 39 ⁣⋯ ⁣5839\!\cdots\!58T8− T^{8} - 11 ⁣⋯ ⁣7211\!\cdots\!72p12T10+ p^{12} T^{10} + 28 ⁣⋯ ⁣8428\!\cdots\!84p24T12−68754072824p36T14+p48T16 p^{24} T^{12} - 68754072824 p^{36} T^{14} + p^{48} T^{16}
61 (1+163738T+5375311858T2−12587109571985600T3−34867093543223492069pT4−12587109571985600p6T5+5375311858p12T6+163738p18T7+p24T8)2 ( 1 + 163738 T + 5375311858 T^{2} - 12587109571985600 T^{3} - 34867093543223492069 p T^{4} - 12587109571985600 p^{6} T^{5} + 5375311858 p^{12} T^{6} + 163738 p^{18} T^{7} + p^{24} T^{8} )^{2}
67 (1−856646T+496161770266T2−190187424030542330T3+ ( 1 - 856646 T + 496161770266 T^{2} - 190187424030542330 T^{3} + 64 ⁣⋯ ⁣2264\!\cdots\!22T4−190187424030542330p6T5+496161770266p12T6−856646p18T7+p24T8)2 T^{4} - 190187424030542330 p^{6} T^{5} + 496161770266 p^{12} T^{6} - 856646 p^{18} T^{7} + p^{24} T^{8} )^{2}
71 1−212021007668T2+ 1 - 212021007668 T^{2} + 13 ⁣⋯ ⁣0413\!\cdots\!04T4− T^{4} - 57 ⁣⋯ ⁣8457\!\cdots\!84T6+ T^{6} + 84 ⁣⋯ ⁣3884\!\cdots\!38T8− T^{8} - 57 ⁣⋯ ⁣8457\!\cdots\!84p12T10+ p^{12} T^{10} + 13 ⁣⋯ ⁣0413\!\cdots\!04p24T12−212021007668p36T14+p48T16 p^{24} T^{12} - 212021007668 p^{36} T^{14} + p^{48} T^{16}
73 (1+1094608T+746233393222T2+336788359861632640T3+ ( 1 + 1094608 T + 746233393222 T^{2} + 336788359861632640 T^{3} + 13 ⁣⋯ ⁣0313\!\cdots\!03T4+336788359861632640p6T5+746233393222p12T6+1094608p18T7+p24T8)2 T^{4} + 336788359861632640 p^{6} T^{5} + 746233393222 p^{12} T^{6} + 1094608 p^{18} T^{7} + p^{24} T^{8} )^{2}
79 (1+8398pT+858780911746T2+463539447756297070T3+ ( 1 + 8398 p T + 858780911746 T^{2} + 463539447756297070 T^{3} + 30 ⁣⋯ ⁣5430\!\cdots\!54T4+463539447756297070p6T5+858780911746p12T6+8398p19T7+p24T8)2 T^{4} + 463539447756297070 p^{6} T^{5} + 858780911746 p^{12} T^{6} + 8398 p^{19} T^{7} + p^{24} T^{8} )^{2}
83 1−1910425105592T2+ 1 - 1910425105592 T^{2} + 16 ⁣⋯ ⁣4416\!\cdots\!44T4− T^{4} - 92 ⁣⋯ ⁣8092\!\cdots\!80T6+ T^{6} + 35 ⁣⋯ ⁣6235\!\cdots\!62T8− T^{8} - 92 ⁣⋯ ⁣8092\!\cdots\!80p12T10+ p^{12} T^{10} + 16 ⁣⋯ ⁣4416\!\cdots\!44p24T12−1910425105592p36T14+p48T16 p^{24} T^{12} - 1910425105592 p^{36} T^{14} + p^{48} T^{16}
89 1−1054207551824T2+ 1 - 1054207551824 T^{2} + 82 ⁣⋯ ⁣7082\!\cdots\!70T4− T^{4} - 51 ⁣⋯ ⁣8451\!\cdots\!84T6+ T^{6} + 24 ⁣⋯ ⁣7524\!\cdots\!75T8− T^{8} - 51 ⁣⋯ ⁣8451\!\cdots\!84p12T10+ p^{12} T^{10} + 82 ⁣⋯ ⁣7082\!\cdots\!70p24T12−1054207551824p36T14+p48T16 p^{24} T^{12} - 1054207551824 p^{36} T^{14} + p^{48} T^{16}
97 (1+1100032T+2005275894904T2+1141210218432366016T3+ ( 1 + 1100032 T + 2005275894904 T^{2} + 1141210218432366016 T^{3} + 17 ⁣⋯ ⁣1817\!\cdots\!18T4+1141210218432366016p6T5+2005275894904p12T6+1100032p18T7+p24T8)2 T^{4} + 1141210218432366016 p^{6} T^{5} + 2005275894904 p^{12} T^{6} + 1100032 p^{18} T^{7} + p^{24} 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

−4.60662838388509470817841173791, −4.39816369798921574547822711741, −4.30667027430073679896242956731, −4.30351383789941592818317926919, −4.14551393925702930196530321284, −4.06052551504018568920692979578, −3.69383235514998664572906287737, −3.66519497439467940766105822609, −3.28751230022939231324379122123, −3.12611661831867952698889134618, −2.96089421146667115527579611197, −2.83662686978823647414834873384, −2.48180395293667665606355664628, −2.37315005162079808384891436105, −2.21280817219030915462233519789, −1.74321159128271946207654877190, −1.73262750793074884281652303882, −1.57479375120172598624867436012, −1.19330644621618535124325887074, −1.19162100874893322881330966975, −1.09952539244804774506001404144, −0.873613194864486893900394988954, −0.46212977243488634455341763186, −0.41285624890765573214059246185, −0.07131499567096626553503045981, 0.07131499567096626553503045981, 0.41285624890765573214059246185, 0.46212977243488634455341763186, 0.873613194864486893900394988954, 1.09952539244804774506001404144, 1.19162100874893322881330966975, 1.19330644621618535124325887074, 1.57479375120172598624867436012, 1.73262750793074884281652303882, 1.74321159128271946207654877190, 2.21280817219030915462233519789, 2.37315005162079808384891436105, 2.48180395293667665606355664628, 2.83662686978823647414834873384, 2.96089421146667115527579611197, 3.12611661831867952698889134618, 3.28751230022939231324379122123, 3.66519497439467940766105822609, 3.69383235514998664572906287737, 4.06052551504018568920692979578, 4.14551393925702930196530321284, 4.30351383789941592818317926919, 4.30667027430073679896242956731, 4.39816369798921574547822711741, 4.60662838388509470817841173791

Graph of the ZZ-function along the critical line

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