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Properties

Label 12-325e6-1.1-c5e6-0-2
Degree 1212
Conductor 1.178×10151.178\times 10^{15}
Sign 11
Analytic cond. 2.00568×10102.00568\times 10^{10}
Root an. cond. 7.219747.21974
Motivic weight 55
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  − 2·2-s − 20·3-s − 27·4-s + 40·6-s − 172·7-s + 124·8-s − 12·9-s + 800·11-s + 540·12-s + 1.01e3·13-s + 344·14-s − 763·16-s − 4.39e3·17-s + 24·18-s + 5.30e3·19-s + 3.44e3·21-s − 1.60e3·22-s + 1.14e3·23-s − 2.48e3·24-s − 2.02e3·26-s + 4.26e3·27-s + 4.64e3·28-s − 3.66e3·29-s + 1.66e4·31-s + 458·32-s − 1.60e4·33-s + 8.79e3·34-s + ⋯
L(s)  = 1  − 0.353·2-s − 1.28·3-s − 0.843·4-s + 0.453·6-s − 1.32·7-s + 0.685·8-s − 0.0493·9-s + 1.99·11-s + 1.08·12-s + 1.66·13-s + 0.469·14-s − 0.745·16-s − 3.68·17-s + 0.0174·18-s + 3.37·19-s + 1.70·21-s − 0.704·22-s + 0.449·23-s − 0.878·24-s − 0.588·26-s + 1.12·27-s + 1.11·28-s − 0.809·29-s + 3.11·31-s + 0.0790·32-s − 2.55·33-s + 1.30·34-s + ⋯

Functional equation

Λ(s)=((512⋅136)s/2 ΓC(s)6 L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
Λ(s)=((512⋅136)s/2 ΓC(s+5/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 512⋅1365^{12} \cdot 13^{6}
Sign: 11
Analytic conductor: 2.00568×10102.00568\times 10^{10}
Root analytic conductor: 7.219747.21974
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 512⋅136, ( :[5/2]6), 1)(12,\ 5^{12} \cdot 13^{6} ,\ ( \ : [5/2]^{6} ),\ 1 )

Particular Values

L(3)L(3) ≈\approx 5.6336602585.633660258
L(12)L(\frac12) ≈\approx 5.6336602585.633660258
L(72)L(\frac{7}{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)
bad5 1 1
13 (1−p2T)6 ( 1 - p^{2} T )^{6}
good2 1+pT+31T2−p3T3+167p3T4−5p6T5+2507p4T6−5p11T7+167p13T8−p18T9+31p20T10+p26T11+p30T12 1 + p T + 31 T^{2} - p^{3} T^{3} + 167 p^{3} T^{4} - 5 p^{6} T^{5} + 2507 p^{4} T^{6} - 5 p^{11} T^{7} + 167 p^{13} T^{8} - p^{18} T^{9} + 31 p^{20} T^{10} + p^{26} T^{11} + p^{30} T^{12}
3 1+20T+412T2+4220T3+6553pT4−94360p2T5−596312p3T6−94360p7T7+6553p11T8+4220p15T9+412p20T10+20p25T11+p30T12 1 + 20 T + 412 T^{2} + 4220 T^{3} + 6553 p T^{4} - 94360 p^{2} T^{5} - 596312 p^{3} T^{6} - 94360 p^{7} T^{7} + 6553 p^{11} T^{8} + 4220 p^{15} T^{9} + 412 p^{20} T^{10} + 20 p^{25} T^{11} + p^{30} T^{12}
7 1+172T+79134T2+10856548T3+409892745pT4+317015702232T5+61415726406116T6+317015702232p5T7+409892745p11T8+10856548p15T9+79134p20T10+172p25T11+p30T12 1 + 172 T + 79134 T^{2} + 10856548 T^{3} + 409892745 p T^{4} + 317015702232 T^{5} + 61415726406116 T^{6} + 317015702232 p^{5} T^{7} + 409892745 p^{11} T^{8} + 10856548 p^{15} T^{9} + 79134 p^{20} T^{10} + 172 p^{25} T^{11} + p^{30} T^{12}
11 1−800T+759568T2−406608064T3+249396661771T4−103203903134944T5+48608265340441376T6−103203903134944p5T7+249396661771p10T8−406608064p15T9+759568p20T10−800p25T11+p30T12 1 - 800 T + 759568 T^{2} - 406608064 T^{3} + 249396661771 T^{4} - 103203903134944 T^{5} + 48608265340441376 T^{6} - 103203903134944 p^{5} T^{7} + 249396661771 p^{10} T^{8} - 406608064 p^{15} T^{9} + 759568 p^{20} T^{10} - 800 p^{25} T^{11} + p^{30} T^{12}
17 1+4396T+13388314T2+30247747580T3+55454091910447T4+84911134688114072T5+ 1 + 4396 T + 13388314 T^{2} + 30247747580 T^{3} + 55454091910447 T^{4} + 84911134688114072 T^{5} + 10 ⁣⋯ ⁣9610\!\cdots\!96T6+84911134688114072p5T7+55454091910447p10T8+30247747580p15T9+13388314p20T10+4396p25T11+p30T12 T^{6} + 84911134688114072 p^{5} T^{7} + 55454091910447 p^{10} T^{8} + 30247747580 p^{15} T^{9} + 13388314 p^{20} T^{10} + 4396 p^{25} T^{11} + p^{30} T^{12}
19 1−5304T+935248pT2−41459051976T3+81939580759323T4−137801885409410896T5+ 1 - 5304 T + 935248 p T^{2} - 41459051976 T^{3} + 81939580759323 T^{4} - 137801885409410896 T^{5} + 22 ⁣⋯ ⁣6822\!\cdots\!68T6−137801885409410896p5T7+81939580759323p10T8−41459051976p15T9+935248p21T10−5304p25T11+p30T12 T^{6} - 137801885409410896 p^{5} T^{7} + 81939580759323 p^{10} T^{8} - 41459051976 p^{15} T^{9} + 935248 p^{21} T^{10} - 5304 p^{25} T^{11} + p^{30} T^{12}
23 1−1140T+26990372T2−27334511148T3+346087829393987T4−321185337113879016T5+ 1 - 1140 T + 26990372 T^{2} - 27334511148 T^{3} + 346087829393987 T^{4} - 321185337113879016 T^{5} + 27 ⁣⋯ ⁣8827\!\cdots\!88T6−321185337113879016p5T7+346087829393987p10T8−27334511148p15T9+26990372p20T10−1140p25T11+p30T12 T^{6} - 321185337113879016 p^{5} T^{7} + 346087829393987 p^{10} T^{8} - 27334511148 p^{15} T^{9} + 26990372 p^{20} T^{10} - 1140 p^{25} T^{11} + p^{30} T^{12}
29 1+3664T+70215246T2+212100467280T3+2582215649047719T4+6374882876649641504T5+ 1 + 3664 T + 70215246 T^{2} + 212100467280 T^{3} + 2582215649047719 T^{4} + 6374882876649641504 T^{5} + 61 ⁣⋯ ⁣4861\!\cdots\!48T6+6374882876649641504p5T7+2582215649047719p10T8+212100467280p15T9+70215246p20T10+3664p25T11+p30T12 T^{6} + 6374882876649641504 p^{5} T^{7} + 2582215649047719 p^{10} T^{8} + 212100467280 p^{15} T^{9} + 70215246 p^{20} T^{10} + 3664 p^{25} T^{11} + p^{30} T^{12}
31 1−16664T+222991608T2−1841948763592T3+13891289359405827T4−80310268298937076688T5+ 1 - 16664 T + 222991608 T^{2} - 1841948763592 T^{3} + 13891289359405827 T^{4} - 80310268298937076688 T^{5} + 47 ⁣⋯ ⁣5647\!\cdots\!56T6−80310268298937076688p5T7+13891289359405827p10T8−1841948763592p15T9+222991608p20T10−16664p25T11+p30T12 T^{6} - 80310268298937076688 p^{5} T^{7} + 13891289359405827 p^{10} T^{8} - 1841948763592 p^{15} T^{9} + 222991608 p^{20} T^{10} - 16664 p^{25} T^{11} + p^{30} T^{12}
37 1+4488T+268756262T2+1050793017672T3+36014260500286007T4+ 1 + 4488 T + 268756262 T^{2} + 1050793017672 T^{3} + 36014260500286007 T^{4} + 11 ⁣⋯ ⁣7611\!\cdots\!76T5+ T^{5} + 30 ⁣⋯ ⁣9630\!\cdots\!96T6+ T^{6} + 11 ⁣⋯ ⁣7611\!\cdots\!76p5T7+36014260500286007p10T8+1050793017672p15T9+268756262p20T10+4488p25T11+p30T12 p^{5} T^{7} + 36014260500286007 p^{10} T^{8} + 1050793017672 p^{15} T^{9} + 268756262 p^{20} T^{10} + 4488 p^{25} T^{11} + p^{30} T^{12}
41 1−12436T+438551018T2−4844551362468T3+106540073505916031T4− 1 - 12436 T + 438551018 T^{2} - 4844551362468 T^{3} + 106540073505916031 T^{4} - 94 ⁣⋯ ⁣0494\!\cdots\!04T5+ T^{5} + 15 ⁣⋯ ⁣9615\!\cdots\!96T6− T^{6} - 94 ⁣⋯ ⁣0494\!\cdots\!04p5T7+106540073505916031p10T8−4844551362468p15T9+438551018p20T10−12436p25T11+p30T12 p^{5} T^{7} + 106540073505916031 p^{10} T^{8} - 4844551362468 p^{15} T^{9} + 438551018 p^{20} T^{10} - 12436 p^{25} T^{11} + p^{30} T^{12}
43 1+1516T+118385308T2+1205424018852T3+22780555260448539T4−19968699827127017192T5+ 1 + 1516 T + 118385308 T^{2} + 1205424018852 T^{3} + 22780555260448539 T^{4} - 19968699827127017192 T^{5} + 44 ⁣⋯ ⁣5644\!\cdots\!56T6−19968699827127017192p5T7+22780555260448539p10T8+1205424018852p15T9+118385308p20T10+1516p25T11+p30T12 T^{6} - 19968699827127017192 p^{5} T^{7} + 22780555260448539 p^{10} T^{8} + 1205424018852 p^{15} T^{9} + 118385308 p^{20} T^{10} + 1516 p^{25} T^{11} + p^{30} T^{12}
47 1+212T+868921678T2−815484002756T3+387087648470244367T4− 1 + 212 T + 868921678 T^{2} - 815484002756 T^{3} + 387087648470244367 T^{4} - 37 ⁣⋯ ⁣7237\!\cdots\!72T5+ T^{5} + 11 ⁣⋯ ⁣8011\!\cdots\!80T6− T^{6} - 37 ⁣⋯ ⁣7237\!\cdots\!72p5T7+387087648470244367p10T8−815484002756p15T9+868921678p20T10+212p25T11+p30T12 p^{5} T^{7} + 387087648470244367 p^{10} T^{8} - 815484002756 p^{15} T^{9} + 868921678 p^{20} T^{10} + 212 p^{25} T^{11} + p^{30} T^{12}
53 1−15612T+1302611762T2−3861503238396T3+464446823633000119T4+ 1 - 15612 T + 1302611762 T^{2} - 3861503238396 T^{3} + 464446823633000119 T^{4} + 86 ⁣⋯ ⁣2486\!\cdots\!24T5+ T^{5} + 79 ⁣⋯ ⁣3679\!\cdots\!36T6+ T^{6} + 86 ⁣⋯ ⁣2486\!\cdots\!24p5T7+464446823633000119p10T8−3861503238396p15T9+1302611762p20T10−15612p25T11+p30T12 p^{5} T^{7} + 464446823633000119 p^{10} T^{8} - 3861503238396 p^{15} T^{9} + 1302611762 p^{20} T^{10} - 15612 p^{25} T^{11} + p^{30} T^{12}
59 1+11896T+2693780976T2+10411916715592T3+3369278181997078251T4+78121391922971510608T5+ 1 + 11896 T + 2693780976 T^{2} + 10411916715592 T^{3} + 3369278181997078251 T^{4} + 78121391922971510608 T^{5} + 28 ⁣⋯ ⁣2428\!\cdots\!24T6+78121391922971510608p5T7+3369278181997078251p10T8+10411916715592p15T9+2693780976p20T10+11896p25T11+p30T12 T^{6} + 78121391922971510608 p^{5} T^{7} + 3369278181997078251 p^{10} T^{8} + 10411916715592 p^{15} T^{9} + 2693780976 p^{20} T^{10} + 11896 p^{25} T^{11} + p^{30} T^{12}
61 1−57232T+5236420718T2−219172905169808T3+11261991887609202663T4− 1 - 57232 T + 5236420718 T^{2} - 219172905169808 T^{3} + 11261991887609202663 T^{4} - 35 ⁣⋯ ⁣0435\!\cdots\!04T5+ T^{5} + 12 ⁣⋯ ⁣2412\!\cdots\!24T6− T^{6} - 35 ⁣⋯ ⁣0435\!\cdots\!04p5T7+11261991887609202663p10T8−219172905169808p15T9+5236420718p20T10−57232p25T11+p30T12 p^{5} T^{7} + 11261991887609202663 p^{10} T^{8} - 219172905169808 p^{15} T^{9} + 5236420718 p^{20} T^{10} - 57232 p^{25} T^{11} + p^{30} T^{12}
67 1−25428T+5004079118T2−103351564549004T3+12139682587774178471T4− 1 - 25428 T + 5004079118 T^{2} - 103351564549004 T^{3} + 12139682587774178471 T^{4} - 19 ⁣⋯ ⁣6019\!\cdots\!60T5+ T^{5} + 19 ⁣⋯ ⁣4419\!\cdots\!44T6− T^{6} - 19 ⁣⋯ ⁣6019\!\cdots\!60p5T7+12139682587774178471p10T8−103351564549004p15T9+5004079118p20T10−25428p25T11+p30T12 p^{5} T^{7} + 12139682587774178471 p^{10} T^{8} - 103351564549004 p^{15} T^{9} + 5004079118 p^{20} T^{10} - 25428 p^{25} T^{11} + p^{30} T^{12}
71 1−214104T+27159407320T2−2418426473306760T3+ 1 - 214104 T + 27159407320 T^{2} - 2418426473306760 T^{3} + 16 ⁣⋯ ⁣9916\!\cdots\!99T4− T^{4} - 94 ⁣⋯ ⁣0094\!\cdots\!00T5+ T^{5} + 44 ⁣⋯ ⁣4844\!\cdots\!48T6− T^{6} - 94 ⁣⋯ ⁣0094\!\cdots\!00p5T7+ p^{5} T^{7} + 16 ⁣⋯ ⁣9916\!\cdots\!99p10T8−2418426473306760p15T9+27159407320p20T10−214104p25T11+p30T12 p^{10} T^{8} - 2418426473306760 p^{15} T^{9} + 27159407320 p^{20} T^{10} - 214104 p^{25} T^{11} + p^{30} T^{12}
73 1+60808T+7683503646T2+466685836081256T3+32200043594087922015T4+ 1 + 60808 T + 7683503646 T^{2} + 466685836081256 T^{3} + 32200043594087922015 T^{4} + 16 ⁣⋯ ⁣2016\!\cdots\!20T5+ T^{5} + 83 ⁣⋯ ⁣8883\!\cdots\!88T6+ T^{6} + 16 ⁣⋯ ⁣2016\!\cdots\!20p5T7+32200043594087922015p10T8+466685836081256p15T9+7683503646p20T10+60808p25T11+p30T12 p^{5} T^{7} + 32200043594087922015 p^{10} T^{8} + 466685836081256 p^{15} T^{9} + 7683503646 p^{20} T^{10} + 60808 p^{25} T^{11} + p^{30} T^{12}
79 1−456pT+11615543106T2−230917448345448T3+62062444497721483599T4− 1 - 456 p T + 11615543106 T^{2} - 230917448345448 T^{3} + 62062444497721483599 T^{4} - 74 ⁣⋯ ⁣4474\!\cdots\!44T5+ T^{5} + 22 ⁣⋯ ⁣6822\!\cdots\!68T6− T^{6} - 74 ⁣⋯ ⁣4474\!\cdots\!44p5T7+62062444497721483599p10T8−230917448345448p15T9+11615543106p20T10−456p26T11+p30T12 p^{5} T^{7} + 62062444497721483599 p^{10} T^{8} - 230917448345448 p^{15} T^{9} + 11615543106 p^{20} T^{10} - 456 p^{26} T^{11} + p^{30} T^{12}
83 1+34684T+7848433574T2+97618429712676T3+32967984989601535079T4− 1 + 34684 T + 7848433574 T^{2} + 97618429712676 T^{3} + 32967984989601535079 T^{4} - 25 ⁣⋯ ⁣4825\!\cdots\!48T5+ T^{5} + 13 ⁣⋯ ⁣4413\!\cdots\!44T6− T^{6} - 25 ⁣⋯ ⁣4825\!\cdots\!48p5T7+32967984989601535079p10T8+97618429712676p15T9+7848433574p20T10+34684p25T11+p30T12 p^{5} T^{7} + 32967984989601535079 p^{10} T^{8} + 97618429712676 p^{15} T^{9} + 7848433574 p^{20} T^{10} + 34684 p^{25} T^{11} + p^{30} T^{12}
89 1+52028T+7379546418T2+426261312231404T3+56427851113243582527T4+ 1 + 52028 T + 7379546418 T^{2} + 426261312231404 T^{3} + 56427851113243582527 T^{4} + 36 ⁣⋯ ⁣6836\!\cdots\!68T5+ T^{5} + 30 ⁣⋯ ⁣0830\!\cdots\!08T6+ T^{6} + 36 ⁣⋯ ⁣6836\!\cdots\!68p5T7+56427851113243582527p10T8+426261312231404p15T9+7379546418p20T10+52028p25T11+p30T12 p^{5} T^{7} + 56427851113243582527 p^{10} T^{8} + 426261312231404 p^{15} T^{9} + 7379546418 p^{20} T^{10} + 52028 p^{25} T^{11} + p^{30} T^{12}
97 1+129316T+26692221250T2+3616210172433684T3+ 1 + 129316 T + 26692221250 T^{2} + 3616210172433684 T^{3} + 39 ⁣⋯ ⁣0739\!\cdots\!07T4+ T^{4} + 43 ⁣⋯ ⁣4043\!\cdots\!40T5+ T^{5} + 41 ⁣⋯ ⁣0441\!\cdots\!04T6+ T^{6} + 43 ⁣⋯ ⁣4043\!\cdots\!40p5T7+ p^{5} T^{7} + 39 ⁣⋯ ⁣0739\!\cdots\!07p10T8+3616210172433684p15T9+26692221250p20T10+129316p25T11+p30T12 p^{10} T^{8} + 3616210172433684 p^{15} T^{9} + 26692221250 p^{20} T^{10} + 129316 p^{25} T^{11} + p^{30} T^{12}
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   L(s)=∏p ∏j=112(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−5.37494558120357542643514262308, −5.07534695243356204283293911182, −4.99031563794642622927811443719, −4.91482120844438773169771813983, −4.81043956800408024057518885081, −4.53421273073423733768006889466, −4.21394190890656730350530220217, −4.03169695043063648149652713051, −3.87562120534043960472924085283, −3.86860504976223299157370362557, −3.47802380528109059262488724646, −3.26609505600707075345612751489, −3.14400769858179058408582426565, −2.82466216104059592068167792476, −2.64595666002383962832282486320, −2.36928963599832644546840189012, −2.12146003094111640263027226297, −1.70407370363802893987987347740, −1.49785837399324914407825110159, −1.45201455924205559157608851914, −0.915255879415186512158254405173, −0.73793836101671324056521906019, −0.52474212714070350780990581765, −0.44491041399355830094882298927, −0.43909575249550549659992038078, 0.43909575249550549659992038078, 0.44491041399355830094882298927, 0.52474212714070350780990581765, 0.73793836101671324056521906019, 0.915255879415186512158254405173, 1.45201455924205559157608851914, 1.49785837399324914407825110159, 1.70407370363802893987987347740, 2.12146003094111640263027226297, 2.36928963599832644546840189012, 2.64595666002383962832282486320, 2.82466216104059592068167792476, 3.14400769858179058408582426565, 3.26609505600707075345612751489, 3.47802380528109059262488724646, 3.86860504976223299157370362557, 3.87562120534043960472924085283, 4.03169695043063648149652713051, 4.21394190890656730350530220217, 4.53421273073423733768006889466, 4.81043956800408024057518885081, 4.91482120844438773169771813983, 4.99031563794642622927811443719, 5.07534695243356204283293911182, 5.37494558120357542643514262308

Graph of the ZZ-function along the critical line

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