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Properties

Label 12-234e6-1.1-c5e6-0-3
Degree 1212
Conductor 1.642×10141.642\times 10^{14}
Sign 11
Analytic cond. 2.79420×1092.79420\times 10^{9}
Root an. cond. 6.126156.12615
Motivic weight 55
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  − 48·4-s + 530·13-s + 1.53e3·16-s + 836·17-s + 416·23-s + 9.73e3·25-s − 1.87e4·29-s − 2.42e4·43-s + 4.89e4·49-s − 2.54e4·52-s + 4.23e4·53-s − 3.19e3·61-s − 4.09e4·64-s − 4.01e4·68-s − 1.69e5·79-s − 1.99e4·92-s − 4.67e5·100-s − 1.38e5·101-s − 3.64e5·103-s + 5.22e4·107-s + 3.13e5·113-s + 9.01e5·116-s + 4.88e5·121-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 3/2·4-s + 0.869·13-s + 3/2·16-s + 0.701·17-s + 0.163·23-s + 3.11·25-s − 4.14·29-s − 1.99·43-s + 2.90·49-s − 1.30·52-s + 2.07·53-s − 0.109·61-s − 5/4·64-s − 1.05·68-s − 3.05·79-s − 0.245·92-s − 4.67·100-s − 1.35·101-s − 3.38·103-s + 0.440·107-s + 2.31·113-s + 6.22·116-s + 3.03·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + ⋯

Functional equation

Λ(s)=((26⋅312⋅136)s/2 ΓC(s)6 L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
Λ(s)=((26⋅312⋅136)s/2 ΓC(s+5/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{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: 26⋅312⋅1362^{6} \cdot 3^{12} \cdot 13^{6}
Sign: 11
Analytic conductor: 2.79420×1092.79420\times 10^{9}
Root analytic conductor: 6.126156.12615
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 26⋅312⋅136, ( :[5/2]6), 1)(12,\ 2^{6} \cdot 3^{12} \cdot 13^{6} ,\ ( \ : [5/2]^{6} ),\ 1 )

Particular Values

L(3)L(3) ≈\approx 6.1179081046.117908104
L(12)L(\frac12) ≈\approx 6.1179081046.117908104
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)
bad2 (1+p4T2)3 ( 1 + p^{4} T^{2} )^{3}
3 1 1
13 1−530T−12785pT2+92996p3T3−12785p6T4−530p10T5+p15T6 1 - 530 T - 12785 p T^{2} + 92996 p^{3} T^{3} - 12785 p^{6} T^{4} - 530 p^{10} T^{5} + p^{15} T^{6}
good5 1−9734T2+45022279T4−151321754836T6+45022279p10T8−9734p20T10+p30T12 1 - 9734 T^{2} + 45022279 T^{4} - 151321754836 T^{6} + 45022279 p^{10} T^{8} - 9734 p^{20} T^{10} + p^{30} T^{12}
7 1−998p2T2+1501179695T4−28969294894196T6+1501179695p10T8−998p22T10+p30T12 1 - 998 p^{2} T^{2} + 1501179695 T^{4} - 28969294894196 T^{6} + 1501179695 p^{10} T^{8} - 998 p^{22} T^{10} + p^{30} T^{12}
11 1−488550T2+122692550487T4−21659528044972276T6+122692550487p10T8−488550p20T10+p30T12 1 - 488550 T^{2} + 122692550487 T^{4} - 21659528044972276 T^{6} + 122692550487 p^{10} T^{8} - 488550 p^{20} T^{10} + p^{30} T^{12}
17 (1−418T+1867887T2+596217860T3+1867887p5T4−418p10T5+p15T6)2 ( 1 - 418 T + 1867887 T^{2} + 596217860 T^{3} + 1867887 p^{5} T^{4} - 418 p^{10} T^{5} + p^{15} T^{6} )^{2}
19 1−4259886T2+15059505195735T4−52704394717503498500T6+15059505195735p10T8−4259886p20T10+p30T12 1 - 4259886 T^{2} + 15059505195735 T^{4} - 52704394717503498500 T^{6} + 15059505195735 p^{10} T^{8} - 4259886 p^{20} T^{10} + p^{30} T^{12}
23 (1−208T+13024485T2−5279785312T3+13024485p5T4−208p10T5+p15T6)2 ( 1 - 208 T + 13024485 T^{2} - 5279785312 T^{3} + 13024485 p^{5} T^{4} - 208 p^{10} T^{5} + p^{15} T^{6} )^{2}
29 (1+9394T+75091011T2+387910039372T3+75091011p5T4+9394p10T5+p15T6)2 ( 1 + 9394 T + 75091011 T^{2} + 387910039372 T^{3} + 75091011 p^{5} T^{4} + 9394 p^{10} T^{5} + p^{15} T^{6} )^{2}
31 1−116335670T2+6009258573309407T4− 1 - 116335670 T^{2} + 6009258573309407 T^{4} - 20 ⁣⋯ ⁣7620\!\cdots\!76T6+6009258573309407p10T8−116335670p20T10+p30T12 T^{6} + 6009258573309407 p^{10} T^{8} - 116335670 p^{20} T^{10} + p^{30} T^{12}
37 1−289018862T2+38604779572315415T4− 1 - 289018862 T^{2} + 38604779572315415 T^{4} - 32 ⁣⋯ ⁣7632\!\cdots\!76T6+38604779572315415p10T8−289018862p20T10+p30T12 T^{6} + 38604779572315415 p^{10} T^{8} - 289018862 p^{20} T^{10} + p^{30} T^{12}
41 1−461990702T2+96326950481430703T4− 1 - 461990702 T^{2} + 96326950481430703 T^{4} - 13 ⁣⋯ ⁣0413\!\cdots\!04T6+96326950481430703p10T8−461990702p20T10+p30T12 T^{6} + 96326950481430703 p^{10} T^{8} - 461990702 p^{20} T^{10} + p^{30} T^{12}
43 (1+12100T+192229169T2+42503254792pT3+192229169p5T4+12100p10T5+p15T6)2 ( 1 + 12100 T + 192229169 T^{2} + 42503254792 p T^{3} + 192229169 p^{5} T^{4} + 12100 p^{10} T^{5} + p^{15} T^{6} )^{2}
47 1−5196642pT2+98471047533657423T4− 1 - 5196642 p T^{2} + 98471047533657423 T^{4} - 23 ⁣⋯ ⁣3223\!\cdots\!32T6+98471047533657423p10T8−5196642p21T10+p30T12 T^{6} + 98471047533657423 p^{10} T^{8} - 5196642 p^{21} T^{10} + p^{30} T^{12}
53 (1−21198T+82289499T2+9230236358604T3+82289499p5T4−21198p10T5+p15T6)2 ( 1 - 21198 T + 82289499 T^{2} + 9230236358604 T^{3} + 82289499 p^{5} T^{4} - 21198 p^{10} T^{5} + p^{15} T^{6} )^{2}
59 1−2269058726T2+2942781419203692343T4− 1 - 2269058726 T^{2} + 2942781419203692343 T^{4} - 25 ⁣⋯ ⁣5625\!\cdots\!56T6+2942781419203692343p10T8−2269058726p20T10+p30T12 T^{6} + 2942781419203692343 p^{10} T^{8} - 2269058726 p^{20} T^{10} + p^{30} T^{12}
61 (1+1598T+927987683T2+20079713107796T3+927987683p5T4+1598p10T5+p15T6)2 ( 1 + 1598 T + 927987683 T^{2} + 20079713107796 T^{3} + 927987683 p^{5} T^{4} + 1598 p^{10} T^{5} + p^{15} T^{6} )^{2}
67 1−3626333582T2+8616065023269477815T4− 1 - 3626333582 T^{2} + 8616065023269477815 T^{4} - 14 ⁣⋯ ⁣7614\!\cdots\!76T6+8616065023269477815p10T8−3626333582p20T10+p30T12 T^{6} + 8616065023269477815 p^{10} T^{8} - 3626333582 p^{20} T^{10} + p^{30} T^{12}
71 1−5654364942T2+254047452086597385pT4− 1 - 5654364942 T^{2} + 254047452086597385 p T^{4} - 37 ⁣⋯ ⁣8837\!\cdots\!88T6+254047452086597385p11T8−5654364942p20T10+p30T12 T^{6} + 254047452086597385 p^{11} T^{8} - 5654364942 p^{20} T^{10} + p^{30} T^{12}
73 1−10335109526T2+47996850402258820223T4− 1 - 10335109526 T^{2} + 47996850402258820223 T^{4} - 12 ⁣⋯ ⁣2812\!\cdots\!28T6+47996850402258820223p10T8−10335109526p20T10+p30T12 T^{6} + 47996850402258820223 p^{10} T^{8} - 10335109526 p^{20} T^{10} + p^{30} T^{12}
79 (1+84664T+9958851181T2+485196394773392T3+9958851181p5T4+84664p10T5+p15T6)2 ( 1 + 84664 T + 9958851181 T^{2} + 485196394773392 T^{3} + 9958851181 p^{5} T^{4} + 84664 p^{10} T^{5} + p^{15} T^{6} )^{2}
83 1−22485370838T2+ 1 - 22485370838 T^{2} + 21 ⁣⋯ ⁣9521\!\cdots\!95T4− T^{4} - 11 ⁣⋯ ⁣0411\!\cdots\!04T6+ T^{6} + 21 ⁣⋯ ⁣9521\!\cdots\!95p10T8−22485370838p20T10+p30T12 p^{10} T^{8} - 22485370838 p^{20} T^{10} + p^{30} T^{12}
89 1−25940313102T2+ 1 - 25940313102 T^{2} + 31 ⁣⋯ ⁣5531\!\cdots\!55T4− T^{4} - 22 ⁣⋯ ⁣6822\!\cdots\!68T6+ T^{6} + 31 ⁣⋯ ⁣5531\!\cdots\!55p10T8−25940313102p20T10+p30T12 p^{10} T^{8} - 25940313102 p^{20} T^{10} + p^{30} T^{12}
97 1−34764784422T2+ 1 - 34764784422 T^{2} + 54 ⁣⋯ ⁣7554\!\cdots\!75T4− T^{4} - 54 ⁣⋯ ⁣1654\!\cdots\!16T6+ T^{6} + 54 ⁣⋯ ⁣7554\!\cdots\!75p10T8−34764784422p20T10+p30T12 p^{10} T^{8} - 34764784422 p^{20} T^{10} + 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

−5.62665225768673021523670576017, −5.56135087788758856848683297164, −5.36552063976034427609362083327, −5.30125081705098699679772420701, −4.87091060317311492091443754455, −4.73139053089582003258059214155, −4.66864683987543627432447143579, −4.26912031861352398738872011437, −4.01227762182539385907210120643, −3.99004334248554720880488743533, −3.84950229464134239014966395960, −3.47281876708367259058623711348, −3.16468289705507836034764247710, −3.13724758851073491568825200625, −3.10334374090735823617379332998, −2.42126937376574835936121129463, −2.38688013680351612716556171628, −2.06720072838183849428592502977, −1.59883081265575553430319927898, −1.54004811445409200498052538396, −1.11967767587726226799828916492, −1.10631504410509656553720295572, −0.50354109323186260384007720550, −0.43622204193111135896320550062, −0.37921355488310683196211631948, 0.37921355488310683196211631948, 0.43622204193111135896320550062, 0.50354109323186260384007720550, 1.10631504410509656553720295572, 1.11967767587726226799828916492, 1.54004811445409200498052538396, 1.59883081265575553430319927898, 2.06720072838183849428592502977, 2.38688013680351612716556171628, 2.42126937376574835936121129463, 3.10334374090735823617379332998, 3.13724758851073491568825200625, 3.16468289705507836034764247710, 3.47281876708367259058623711348, 3.84950229464134239014966395960, 3.99004334248554720880488743533, 4.01227762182539385907210120643, 4.26912031861352398738872011437, 4.66864683987543627432447143579, 4.73139053089582003258059214155, 4.87091060317311492091443754455, 5.30125081705098699679772420701, 5.36552063976034427609362083327, 5.56135087788758856848683297164, 5.62665225768673021523670576017

Graph of the ZZ-function along the critical line

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