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Properties

Label 14-1232e7-1.1-c3e7-0-1
Degree 1414
Conductor 4.308×10214.308\times 10^{21}
Sign 11
Analytic cond. 1.07235×10131.07235\times 10^{13}
Root an. cond. 8.525868.52586
Motivic weight 33
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  + 6·5-s + 49·7-s − 44·9-s + 77·11-s + 88·13-s + 134·17-s − 14·19-s − 42·23-s − 248·25-s + 10·27-s + 482·29-s − 50·31-s + 294·35-s + 152·37-s + 234·41-s − 472·43-s − 264·45-s − 728·47-s + 1.37e3·49-s − 102·53-s + 462·55-s − 1.70e3·59-s + 656·61-s − 2.15e3·63-s + 528·65-s − 1.12e3·67-s + 918·71-s + ⋯
L(s)  = 1  + 0.536·5-s + 2.64·7-s − 1.62·9-s + 2.11·11-s + 1.87·13-s + 1.91·17-s − 0.169·19-s − 0.380·23-s − 1.98·25-s + 0.0712·27-s + 3.08·29-s − 0.289·31-s + 1.41·35-s + 0.675·37-s + 0.891·41-s − 1.67·43-s − 0.874·45-s − 2.25·47-s + 4·49-s − 0.264·53-s + 1.13·55-s − 3.76·59-s + 1.37·61-s − 4.31·63-s + 1.00·65-s − 2.05·67-s + 1.53·71-s + ⋯

Functional equation

Λ(s)=((228⋅77⋅117)s/2 ΓC(s)7 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 7^{7} \cdot 11^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((228⋅77⋅117)s/2 ΓC(s+3/2)7 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 7^{7} \cdot 11^{7}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1414
Conductor: 228⋅77⋅1172^{28} \cdot 7^{7} \cdot 11^{7}
Sign: 11
Analytic conductor: 1.07235×10131.07235\times 10^{13}
Root analytic conductor: 8.525868.52586
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (14, 228⋅77⋅117, ( :[3/2]7), 1)(14,\ 2^{28} \cdot 7^{7} \cdot 11^{7} ,\ ( \ : [3/2]^{7} ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 61.4230111761.42301117
L(12)L(\frac12) ≈\approx 61.4230111761.42301117
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)
bad2 1 1
7 (1−pT)7 ( 1 - p T )^{7}
11 (1−pT)7 ( 1 - p T )^{7}
good3 1+44T2−10T3+524T4−3532T5+18349T6−175828T7+18349p3T8−3532p6T9+524p9T10−10p12T11+44p15T12+p21T14 1 + 44 T^{2} - 10 T^{3} + 524 T^{4} - 3532 T^{5} + 18349 T^{6} - 175828 T^{7} + 18349 p^{3} T^{8} - 3532 p^{6} T^{9} + 524 p^{9} T^{10} - 10 p^{12} T^{11} + 44 p^{15} T^{12} + p^{21} T^{14}
5 1−6T+284T2−2272T3+65826T4−404002T5+2026369pT6−67987536T7+2026369p4T8−404002p6T9+65826p9T10−2272p12T11+284p15T12−6p18T13+p21T14 1 - 6 T + 284 T^{2} - 2272 T^{3} + 65826 T^{4} - 404002 T^{5} + 2026369 p T^{6} - 67987536 T^{7} + 2026369 p^{4} T^{8} - 404002 p^{6} T^{9} + 65826 p^{9} T^{10} - 2272 p^{12} T^{11} + 284 p^{15} T^{12} - 6 p^{18} T^{13} + p^{21} T^{14}
13 1−88T+9809T2−715488T3+54045339T4−3006894808T5+175980233971T6−8250801499552T7+175980233971p3T8−3006894808p6T9+54045339p9T10−715488p12T11+9809p15T12−88p18T13+p21T14 1 - 88 T + 9809 T^{2} - 715488 T^{3} + 54045339 T^{4} - 3006894808 T^{5} + 175980233971 T^{6} - 8250801499552 T^{7} + 175980233971 p^{3} T^{8} - 3006894808 p^{6} T^{9} + 54045339 p^{9} T^{10} - 715488 p^{12} T^{11} + 9809 p^{15} T^{12} - 88 p^{18} T^{13} + p^{21} T^{14}
17 1−134T+27305T2−2485848T3+290356439T4−20303562618T5+1833806643903T6−111127946382800T7+1833806643903p3T8−20303562618p6T9+290356439p9T10−2485848p12T11+27305p15T12−134p18T13+p21T14 1 - 134 T + 27305 T^{2} - 2485848 T^{3} + 290356439 T^{4} - 20303562618 T^{5} + 1833806643903 T^{6} - 111127946382800 T^{7} + 1833806643903 p^{3} T^{8} - 20303562618 p^{6} T^{9} + 290356439 p^{9} T^{10} - 2485848 p^{12} T^{11} + 27305 p^{15} T^{12} - 134 p^{18} T^{13} + p^{21} T^{14}
19 1+14T+15445T2+370116T3+221820981T4+3362375266T5+1940890827609T6+36078670195896T7+1940890827609p3T8+3362375266p6T9+221820981p9T10+370116p12T11+15445p15T12+14p18T13+p21T14 1 + 14 T + 15445 T^{2} + 370116 T^{3} + 221820981 T^{4} + 3362375266 T^{5} + 1940890827609 T^{6} + 36078670195896 T^{7} + 1940890827609 p^{3} T^{8} + 3362375266 p^{6} T^{9} + 221820981 p^{9} T^{10} + 370116 p^{12} T^{11} + 15445 p^{15} T^{12} + 14 p^{18} T^{13} + p^{21} T^{14}
23 1+42T+27842T2−501852T3+488798080T4−10372070314T5+8502794586935T6−72704581429128T7+8502794586935p3T8−10372070314p6T9+488798080p9T10−501852p12T11+27842p15T12+42p18T13+p21T14 1 + 42 T + 27842 T^{2} - 501852 T^{3} + 488798080 T^{4} - 10372070314 T^{5} + 8502794586935 T^{6} - 72704581429128 T^{7} + 8502794586935 p^{3} T^{8} - 10372070314 p^{6} T^{9} + 488798080 p^{9} T^{10} - 501852 p^{12} T^{11} + 27842 p^{15} T^{12} + 42 p^{18} T^{13} + p^{21} T^{14}
29 1−482T+228623T2−69234996T3+19038406729T4−4147230797646T5+814089784958751T6−133910789584078360T7+814089784958751p3T8−4147230797646p6T9+19038406729p9T10−69234996p12T11+228623p15T12−482p18T13+p21T14 1 - 482 T + 228623 T^{2} - 69234996 T^{3} + 19038406729 T^{4} - 4147230797646 T^{5} + 814089784958751 T^{6} - 133910789584078360 T^{7} + 814089784958751 p^{3} T^{8} - 4147230797646 p^{6} T^{9} + 19038406729 p^{9} T^{10} - 69234996 p^{12} T^{11} + 228623 p^{15} T^{12} - 482 p^{18} T^{13} + p^{21} T^{14}
31 1+50T+165036T2+11906720T3+12248821132T4+1007923922542T5+17662703433551pT6+41569550168789568T7+17662703433551p4T8+1007923922542p6T9+12248821132p9T10+11906720p12T11+165036p15T12+50p18T13+p21T14 1 + 50 T + 165036 T^{2} + 11906720 T^{3} + 12248821132 T^{4} + 1007923922542 T^{5} + 17662703433551 p T^{6} + 41569550168789568 T^{7} + 17662703433551 p^{4} T^{8} + 1007923922542 p^{6} T^{9} + 12248821132 p^{9} T^{10} + 11906720 p^{12} T^{11} + 165036 p^{15} T^{12} + 50 p^{18} T^{13} + p^{21} T^{14}
37 1−152T+48644T2+3370374T3+914590226T4+238147307416T5+231060901239309T6−34349523495575916T7+231060901239309p3T8+238147307416p6T9+914590226p9T10+3370374p12T11+48644p15T12−152p18T13+p21T14 1 - 152 T + 48644 T^{2} + 3370374 T^{3} + 914590226 T^{4} + 238147307416 T^{5} + 231060901239309 T^{6} - 34349523495575916 T^{7} + 231060901239309 p^{3} T^{8} + 238147307416 p^{6} T^{9} + 914590226 p^{9} T^{10} + 3370374 p^{12} T^{11} + 48644 p^{15} T^{12} - 152 p^{18} T^{13} + p^{21} T^{14}
41 1−234T+5961pT2−32812360T3+26027650695T4−2322405515318T5+2098716471450151T6−166131275752043120T7+2098716471450151p3T8−2322405515318p6T9+26027650695p9T10−32812360p12T11+5961p16T12−234p18T13+p21T14 1 - 234 T + 5961 p T^{2} - 32812360 T^{3} + 26027650695 T^{4} - 2322405515318 T^{5} + 2098716471450151 T^{6} - 166131275752043120 T^{7} + 2098716471450151 p^{3} T^{8} - 2322405515318 p^{6} T^{9} + 26027650695 p^{9} T^{10} - 32812360 p^{12} T^{11} + 5961 p^{16} T^{12} - 234 p^{18} T^{13} + p^{21} T^{14}
43 1+472T+299249T2+141073184T3+54709149297T4+20732706560040T5+6662097845348473T6+1954933217456521792T7+6662097845348473p3T8+20732706560040p6T9+54709149297p9T10+141073184p12T11+299249p15T12+472p18T13+p21T14 1 + 472 T + 299249 T^{2} + 141073184 T^{3} + 54709149297 T^{4} + 20732706560040 T^{5} + 6662097845348473 T^{6} + 1954933217456521792 T^{7} + 6662097845348473 p^{3} T^{8} + 20732706560040 p^{6} T^{9} + 54709149297 p^{9} T^{10} + 141073184 p^{12} T^{11} + 299249 p^{15} T^{12} + 472 p^{18} T^{13} + p^{21} T^{14}
47 1+728T+531007T2+279637504T3+137101200639T4+56205275522408T5+21393386805952065T6+7209517394974785920T7+21393386805952065p3T8+56205275522408p6T9+137101200639p9T10+279637504p12T11+531007p15T12+728p18T13+p21T14 1 + 728 T + 531007 T^{2} + 279637504 T^{3} + 137101200639 T^{4} + 56205275522408 T^{5} + 21393386805952065 T^{6} + 7209517394974785920 T^{7} + 21393386805952065 p^{3} T^{8} + 56205275522408 p^{6} T^{9} + 137101200639 p^{9} T^{10} + 279637504 p^{12} T^{11} + 531007 p^{15} T^{12} + 728 p^{18} T^{13} + p^{21} T^{14}
53 1+102T+362147T2+58937924T3+66192367093T4+16550013323178T5+11223024004883295T6+3035593503241377848T7+11223024004883295p3T8+16550013323178p6T9+66192367093p9T10+58937924p12T11+362147p15T12+102p18T13+p21T14 1 + 102 T + 362147 T^{2} + 58937924 T^{3} + 66192367093 T^{4} + 16550013323178 T^{5} + 11223024004883295 T^{6} + 3035593503241377848 T^{7} + 11223024004883295 p^{3} T^{8} + 16550013323178 p^{6} T^{9} + 66192367093 p^{9} T^{10} + 58937924 p^{12} T^{11} + 362147 p^{15} T^{12} + 102 p^{18} T^{13} + p^{21} T^{14}
59 1+1704T+1862148T2+1368305002T3+874437219132T4+491863769450180T5+4551080320375079pT6+ 1 + 1704 T + 1862148 T^{2} + 1368305002 T^{3} + 874437219132 T^{4} + 491863769450180 T^{5} + 4551080320375079 p T^{6} + 12 ⁣⋯ ⁣6412\!\cdots\!64T7+4551080320375079p4T8+491863769450180p6T9+874437219132p9T10+1368305002p12T11+1862148p15T12+1704p18T13+p21T14 T^{7} + 4551080320375079 p^{4} T^{8} + 491863769450180 p^{6} T^{9} + 874437219132 p^{9} T^{10} + 1368305002 p^{12} T^{11} + 1862148 p^{15} T^{12} + 1704 p^{18} T^{13} + p^{21} T^{14}
61 1−656T+1183501T2−579578048T3+602414873903T4−232960751490176T5+187633183287067091T6−61372942309294973152T7+187633183287067091p3T8−232960751490176p6T9+602414873903p9T10−579578048p12T11+1183501p15T12−656p18T13+p21T14 1 - 656 T + 1183501 T^{2} - 579578048 T^{3} + 602414873903 T^{4} - 232960751490176 T^{5} + 187633183287067091 T^{6} - 61372942309294973152 T^{7} + 187633183287067091 p^{3} T^{8} - 232960751490176 p^{6} T^{9} + 602414873903 p^{9} T^{10} - 579578048 p^{12} T^{11} + 1183501 p^{15} T^{12} - 656 p^{18} T^{13} + p^{21} T^{14}
67 1+1126T+1986742T2+1557736324T3+24148623212pT4+983663348300682T5+757552257136066331T6+ 1 + 1126 T + 1986742 T^{2} + 1557736324 T^{3} + 24148623212 p T^{4} + 983663348300682 T^{5} + 757552257136066331 T^{6} + 37 ⁣⋯ ⁣2037\!\cdots\!20T7+757552257136066331p3T8+983663348300682p6T9+24148623212p10T10+1557736324p12T11+1986742p15T12+1126p18T13+p21T14 T^{7} + 757552257136066331 p^{3} T^{8} + 983663348300682 p^{6} T^{9} + 24148623212 p^{10} T^{10} + 1557736324 p^{12} T^{11} + 1986742 p^{15} T^{12} + 1126 p^{18} T^{13} + p^{21} T^{14}
71 1−918T+1410226T2−1127188880T3+1024999468960T4−707246195335354T5+489168566062675767T6− 1 - 918 T + 1410226 T^{2} - 1127188880 T^{3} + 1024999468960 T^{4} - 707246195335354 T^{5} + 489168566062675767 T^{6} - 30 ⁣⋯ ⁣8830\!\cdots\!88T7+489168566062675767p3T8−707246195335354p6T9+1024999468960p9T10−1127188880p12T11+1410226p15T12−918p18T13+p21T14 T^{7} + 489168566062675767 p^{3} T^{8} - 707246195335354 p^{6} T^{9} + 1024999468960 p^{9} T^{10} - 1127188880 p^{12} T^{11} + 1410226 p^{15} T^{12} - 918 p^{18} T^{13} + p^{21} T^{14}
73 1−1094T+956505T2+29360648T3−154335549345T4+138405554077542T5+169049507742265063T6− 1 - 1094 T + 956505 T^{2} + 29360648 T^{3} - 154335549345 T^{4} + 138405554077542 T^{5} + 169049507742265063 T^{6} - 12 ⁣⋯ ⁣0012\!\cdots\!00T7+169049507742265063p3T8+138405554077542p6T9−154335549345p9T10+29360648p12T11+956505p15T12−1094p18T13+p21T14 T^{7} + 169049507742265063 p^{3} T^{8} + 138405554077542 p^{6} T^{9} - 154335549345 p^{9} T^{10} + 29360648 p^{12} T^{11} + 956505 p^{15} T^{12} - 1094 p^{18} T^{13} + p^{21} T^{14}
79 1+672T+1991289T2+1005883200T3+1879401075445T4+854908116670688T5+1258365779798928861T6+ 1 + 672 T + 1991289 T^{2} + 1005883200 T^{3} + 1879401075445 T^{4} + 854908116670688 T^{5} + 1258365779798928861 T^{6} + 51 ⁣⋯ ⁣8851\!\cdots\!88T7+1258365779798928861p3T8+854908116670688p6T9+1879401075445p9T10+1005883200p12T11+1991289p15T12+672p18T13+p21T14 T^{7} + 1258365779798928861 p^{3} T^{8} + 854908116670688 p^{6} T^{9} + 1879401075445 p^{9} T^{10} + 1005883200 p^{12} T^{11} + 1991289 p^{15} T^{12} + 672 p^{18} T^{13} + p^{21} T^{14}
83 1+782T+2973761T2+1635268700T3+3819085657785T4+1534814206273218T5+3016850158169180577T6+ 1 + 782 T + 2973761 T^{2} + 1635268700 T^{3} + 3819085657785 T^{4} + 1534814206273218 T^{5} + 3016850158169180577 T^{6} + 97 ⁣⋯ ⁣6497\!\cdots\!64T7+3016850158169180577p3T8+1534814206273218p6T9+3819085657785p9T10+1635268700p12T11+2973761p15T12+782p18T13+p21T14 T^{7} + 3016850158169180577 p^{3} T^{8} + 1534814206273218 p^{6} T^{9} + 3819085657785 p^{9} T^{10} + 1635268700 p^{12} T^{11} + 2973761 p^{15} T^{12} + 782 p^{18} T^{13} + p^{21} T^{14}
89 1+464T+2174152T2+844413518T3+2524441633154T4+912233491401400T5+2197789561954212749T6+ 1 + 464 T + 2174152 T^{2} + 844413518 T^{3} + 2524441633154 T^{4} + 912233491401400 T^{5} + 2197789561954212749 T^{6} + 78 ⁣⋯ ⁣2078\!\cdots\!20T7+2197789561954212749p3T8+912233491401400p6T9+2524441633154p9T10+844413518p12T11+2174152p15T12+464p18T13+p21T14 T^{7} + 2197789561954212749 p^{3} T^{8} + 912233491401400 p^{6} T^{9} + 2524441633154 p^{9} T^{10} + 844413518 p^{12} T^{11} + 2174152 p^{15} T^{12} + 464 p^{18} T^{13} + p^{21} T^{14}
97 1−3000T+6069912T2−9347999414T3+12452984432090T4−14784931980343936T5+16273667759607265133T6− 1 - 3000 T + 6069912 T^{2} - 9347999414 T^{3} + 12452984432090 T^{4} - 14784931980343936 T^{5} + 16273667759607265133 T^{6} - 16 ⁣⋯ ⁣4016\!\cdots\!40T7+16273667759607265133p3T8−14784931980343936p6T9+12452984432090p9T10−9347999414p12T11+6069912p15T12−3000p18T13+p21T14 T^{7} + 16273667759607265133 p^{3} T^{8} - 14784931980343936 p^{6} T^{9} + 12452984432090 p^{9} T^{10} - 9347999414 p^{12} T^{11} + 6069912 p^{15} T^{12} - 3000 p^{18} T^{13} + p^{21} T^{14}
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   L(s)=∏p ∏j=114(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−4.26152928310261404084630890402, −3.94382895927978185697756710197, −3.93975732662952024106181305291, −3.82314871952104351976285773287, −3.40037582810331154894625330305, −3.35437390901574907664249860263, −3.18740671825920417364786833311, −3.14577510364710213365497718125, −3.09908034623271819435335173175, −3.08409610056267059535333611063, −2.78723704626474249035089916959, −2.21802647403040074042824207171, −2.17231899093746881109069787923, −2.12858825219902819317042048998, −1.94516317992995127595829630777, −1.85791546008814427942007511280, −1.66143365173285989362741250358, −1.55151143276967361652105575604, −1.34998767291338910683915493150, −1.02092803072629019538741852465, −0.927199658352570073717193443460, −0.918957780529917934633803546381, −0.65289168032938232957733969051, −0.36606981098494498423112312431, −0.35465166173679589279114760821, 0.35465166173679589279114760821, 0.36606981098494498423112312431, 0.65289168032938232957733969051, 0.918957780529917934633803546381, 0.927199658352570073717193443460, 1.02092803072629019538741852465, 1.34998767291338910683915493150, 1.55151143276967361652105575604, 1.66143365173285989362741250358, 1.85791546008814427942007511280, 1.94516317992995127595829630777, 2.12858825219902819317042048998, 2.17231899093746881109069787923, 2.21802647403040074042824207171, 2.78723704626474249035089916959, 3.08409610056267059535333611063, 3.09908034623271819435335173175, 3.14577510364710213365497718125, 3.18740671825920417364786833311, 3.35437390901574907664249860263, 3.40037582810331154894625330305, 3.82314871952104351976285773287, 3.93975732662952024106181305291, 3.94382895927978185697756710197, 4.26152928310261404084630890402

Graph of the ZZ-function along the critical line

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