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Properties

Label 12-2240e6-1.1-c1e6-0-1
Degree 1212
Conductor 1.263×10201.263\times 10^{20}
Sign 11
Analytic cond. 3.27454×1073.27454\times 10^{7}
Root an. cond. 4.229244.22924
Motivic weight 11
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  + 5·9-s − 14·11-s + 8·19-s − 5·25-s + 6·29-s + 20·31-s + 36·41-s − 3·49-s + 12·59-s − 48·61-s + 8·71-s − 34·79-s + 14·81-s − 70·99-s + 8·101-s + 10·109-s + 65·121-s + 8·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 5/3·9-s − 4.22·11-s + 1.83·19-s − 25-s + 1.11·29-s + 3.59·31-s + 5.62·41-s − 3/7·49-s + 1.56·59-s − 6.14·61-s + 0.949·71-s − 3.82·79-s + 14/9·81-s − 7.03·99-s + 0.796·101-s + 0.957·109-s + 5.90·121-s + 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

Λ(s)=((236⋅56⋅76)s/2 ΓC(s)6 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((236⋅56⋅76)s/2 ΓC(s+1/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 236⋅56⋅762^{36} \cdot 5^{6} \cdot 7^{6}
Sign: 11
Analytic conductor: 3.27454×1073.27454\times 10^{7}
Root analytic conductor: 4.229244.22924
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 236⋅56⋅76, ( :[1/2]6), 1)(12,\ 2^{36} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.1443773631.144377363
L(12)L(\frac12) ≈\approx 1.1443773631.144377363
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+pT2−8T3+p2T4+p3T6 1 + p T^{2} - 8 T^{3} + p^{2} T^{4} + p^{3} T^{6}
7 (1+T2)3 ( 1 + T^{2} )^{3}
good3 1−5T2+11T4−26T6+11p2T8−5p4T10+p6T12 1 - 5 T^{2} + 11 T^{4} - 26 T^{6} + 11 p^{2} T^{8} - 5 p^{4} T^{10} + p^{6} T^{12}
11 (1+7T+41T2+146T3+41pT4+7p2T5+p3T6)2 ( 1 + 7 T + 41 T^{2} + 146 T^{3} + 41 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2}
13 1−9T2+491T4−2882T6+491p2T8−9p4T10+p6T12 1 - 9 T^{2} + 491 T^{4} - 2882 T^{6} + 491 p^{2} T^{8} - 9 p^{4} T^{10} + p^{6} T^{12}
17 1−53T2+1539T4−31118T6+1539p2T8−53p4T10+p6T12 1 - 53 T^{2} + 1539 T^{4} - 31118 T^{6} + 1539 p^{2} T^{8} - 53 p^{4} T^{10} + p^{6} T^{12}
19 (1−4T+43T2−160T3+43pT4−4p2T5+p3T6)2 ( 1 - 4 T + 43 T^{2} - 160 T^{3} + 43 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2}
23 1−2pT2+1887T4−43972T6+1887p2T8−2p5T10+p6T12 1 - 2 p T^{2} + 1887 T^{4} - 43972 T^{6} + 1887 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12}
29 (1−3T+15T2−66T3+15pT4−3p2T5+p3T6)2 ( 1 - 3 T + 15 T^{2} - 66 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2}
31 (1−10T+101T2−540T3+101pT4−10p2T5+p3T6)2 ( 1 - 10 T + 101 T^{2} - 540 T^{3} + 101 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2}
37 (1−38T2+p2T4)3 ( 1 - 38 T^{2} + p^{2} T^{4} )^{3}
41 (1−18T+191T2−1388T3+191pT4−18p2T5+p3T6)2 ( 1 - 18 T + 191 T^{2} - 1388 T^{3} + 191 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2}
43 1−178T2+15575T4−836124T6+15575p2T8−178p4T10+p6T12 1 - 178 T^{2} + 15575 T^{4} - 836124 T^{6} + 15575 p^{2} T^{8} - 178 p^{4} T^{10} + p^{6} T^{12}
47 1−105T2+6339T4−285798T6+6339p2T8−105p4T10+p6T12 1 - 105 T^{2} + 6339 T^{4} - 285798 T^{6} + 6339 p^{2} T^{8} - 105 p^{4} T^{10} + p^{6} T^{12}
53 1−130T2+13655T4−792060T6+13655p2T8−130p4T10+p6T12 1 - 130 T^{2} + 13655 T^{4} - 792060 T^{6} + 13655 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12}
59 (1−6T+99T2−752T3+99pT4−6p2T5+p3T6)2 ( 1 - 6 T + 99 T^{2} - 752 T^{3} + 99 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2}
61 (1+24T+365T2+3368T3+365pT4+24p2T5+p3T6)2 ( 1 + 24 T + 365 T^{2} + 3368 T^{3} + 365 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} )^{2}
67 1−174T2+20567T4−1533188T6+20567p2T8−174p4T10+p6T12 1 - 174 T^{2} + 20567 T^{4} - 1533188 T^{6} + 20567 p^{2} T^{8} - 174 p^{4} T^{10} + p^{6} T^{12}
71 (1−4T+193T2−504T3+193pT4−4p2T5+p3T6)2 ( 1 - 4 T + 193 T^{2} - 504 T^{3} + 193 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2}
73 1−346T2+55487T4−5172972T6+55487p2T8−346p4T10+p6T12 1 - 346 T^{2} + 55487 T^{4} - 5172972 T^{6} + 55487 p^{2} T^{8} - 346 p^{4} T^{10} + p^{6} T^{12}
79 (1+17T+205T2+2138T3+205pT4+17p2T5+p3T6)2 ( 1 + 17 T + 205 T^{2} + 2138 T^{3} + 205 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} )^{2}
83 1−70T2+1179T4+304148T6+1179p2T8−70p4T10+p6T12 1 - 70 T^{2} + 1179 T^{4} + 304148 T^{6} + 1179 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12}
89 (1+95T2+464T3+95pT4+p3T6)2 ( 1 + 95 T^{2} + 464 T^{3} + 95 p T^{4} + p^{3} T^{6} )^{2}
97 1−469T2+98339T4−12075534T6+98339p2T8−469p4T10+p6T12 1 - 469 T^{2} + 98339 T^{4} - 12075534 T^{6} + 98339 p^{2} T^{8} - 469 p^{4} T^{10} + p^{6} 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

−4.69815035448418291999517402964, −4.51098445734869472639692642442, −4.37997368707955189165282506806, −4.36723065633882974752625138575, −4.23762783412656626156825203807, −4.10487345572472373686447857700, −3.78121504769123861144496794781, −3.72886529994956842613722420228, −3.29741240262584375005363382817, −3.14617134104138740823008942077, −3.00332364357698378627119875471, −2.90955795261554342058630492544, −2.89715030398171804771310048878, −2.63535842161519751676928039204, −2.60194272619800707753019422572, −2.35222366684003505636232296208, −2.10350572550630194825112209802, −2.08956047455687522791673897413, −1.69937272077775759384195679956, −1.29342535865369396081798939881, −1.27035579803350239097332096092, −1.11383796772185616752951366813, −0.71964500885534532917531131402, −0.64255595511844002602397299692, −0.12339308480590164766582464196, 0.12339308480590164766582464196, 0.64255595511844002602397299692, 0.71964500885534532917531131402, 1.11383796772185616752951366813, 1.27035579803350239097332096092, 1.29342535865369396081798939881, 1.69937272077775759384195679956, 2.08956047455687522791673897413, 2.10350572550630194825112209802, 2.35222366684003505636232296208, 2.60194272619800707753019422572, 2.63535842161519751676928039204, 2.89715030398171804771310048878, 2.90955795261554342058630492544, 3.00332364357698378627119875471, 3.14617134104138740823008942077, 3.29741240262584375005363382817, 3.72886529994956842613722420228, 3.78121504769123861144496794781, 4.10487345572472373686447857700, 4.23762783412656626156825203807, 4.36723065633882974752625138575, 4.37997368707955189165282506806, 4.51098445734869472639692642442, 4.69815035448418291999517402964

Graph of the ZZ-function along the critical line

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