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Properties

Label 12-135e6-1.1-c1e6-0-0
Degree $12$
Conductor $6.053\times 10^{12}$
Sign $1$
Analytic cond. $1.56915$
Root an. cond. $1.03825$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 3·5-s − 5·7-s + 3·10-s − 2·11-s − 4·13-s − 5·14-s + 16-s + 4·17-s + 8·19-s + 3·20-s − 2·22-s + 3·23-s + 3·25-s − 4·26-s − 5·28-s − 7·29-s − 8·31-s + 3·32-s + 4·34-s − 15·35-s + 12·37-s + 8·38-s − 13·41-s − 10·43-s − 2·44-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.34·5-s − 1.88·7-s + 0.948·10-s − 0.603·11-s − 1.10·13-s − 1.33·14-s + 1/4·16-s + 0.970·17-s + 1.83·19-s + 0.670·20-s − 0.426·22-s + 0.625·23-s + 3/5·25-s − 0.784·26-s − 0.944·28-s − 1.29·29-s − 1.43·31-s + 0.530·32-s + 0.685·34-s − 2.53·35-s + 1.97·37-s + 1.29·38-s − 2.03·41-s − 1.52·43-s − 0.301·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{18} \cdot 5^{6}\)
Sign: $1$
Analytic conductor: \(1.56915\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{18} \cdot 5^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.805133844\)
\(L(\frac12)\) \(\approx\) \(1.805133844\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( ( 1 - T + T^{2} )^{3} \)
good2 \( 1 - T + T^{3} - p T^{4} - T^{5} + 11 T^{6} - p T^{7} - p^{3} T^{8} + p^{3} T^{9} - p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 5 T + T^{2} - 2 p T^{3} + 73 T^{4} + 173 T^{5} - 82 T^{6} + 173 p T^{7} + 73 p^{2} T^{8} - 2 p^{4} T^{9} + p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 2 T - 21 T^{2} - 14 T^{3} + 26 p T^{4} - 58 T^{5} - 3673 T^{6} - 58 p T^{7} + 26 p^{3} T^{8} - 14 p^{3} T^{9} - 21 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 4 T - 19 T^{2} - 60 T^{3} + 370 T^{4} + 536 T^{5} - 4235 T^{6} + 536 p T^{7} + 370 p^{2} T^{8} - 60 p^{3} T^{9} - 19 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
17 \( ( 1 - 2 T + 43 T^{2} - 56 T^{3} + 43 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( ( 1 - 4 T + 53 T^{2} - 148 T^{3} + 53 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 3 T - 27 T^{2} - 66 T^{3} + 405 T^{4} + 2625 T^{5} - 9794 T^{6} + 2625 p T^{7} + 405 p^{2} T^{8} - 66 p^{3} T^{9} - 27 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 7 T - 9 T^{2} - 304 T^{3} - 803 T^{4} + 101 p T^{5} + 36038 T^{6} + 101 p^{2} T^{7} - 803 p^{2} T^{8} - 304 p^{3} T^{9} - 9 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 8 T + p T^{2} + 208 T^{3} - 158 T^{4} - 7756 T^{5} - 34897 T^{6} - 7756 p T^{7} - 158 p^{2} T^{8} + 208 p^{3} T^{9} + p^{5} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 - 6 T + 99 T^{2} - 440 T^{3} + 99 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 13 T + 27 T^{2} - 292 T^{3} + 445 T^{4} + 22279 T^{5} + 169790 T^{6} + 22279 p T^{7} + 445 p^{2} T^{8} - 292 p^{3} T^{9} + 27 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 10 T - 25 T^{2} - 462 T^{3} + 1690 T^{4} + 17990 T^{5} + 34975 T^{6} + 17990 p T^{7} + 1690 p^{2} T^{8} - 462 p^{3} T^{9} - 25 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 13 T + 39 T^{2} + 16 T^{3} - 299 T^{4} + 19253 T^{5} - 232366 T^{6} + 19253 p T^{7} - 299 p^{2} T^{8} + 16 p^{3} T^{9} + 39 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 - 2 T + 139 T^{2} - 188 T^{3} + 139 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 2 T - 153 T^{2} - 110 T^{3} + 14962 T^{4} + 3194 T^{5} - 1012513 T^{6} + 3194 p T^{7} + 14962 p^{2} T^{8} - 110 p^{3} T^{9} - 153 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + T - 145 T^{2} - 240 T^{3} + 12217 T^{4} + 14087 T^{5} - 812786 T^{6} + 14087 p T^{7} + 12217 p^{2} T^{8} - 240 p^{3} T^{9} - 145 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 11 T - 41 T^{2} - 152 T^{3} + 4753 T^{4} - 32755 T^{5} - 764902 T^{6} - 32755 p T^{7} + 4753 p^{2} T^{8} - 152 p^{3} T^{9} - 41 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 10 T + 121 T^{2} - 712 T^{3} + 121 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 + 8 T + 155 T^{2} + 1040 T^{3} + 155 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 + 2 T - 149 T^{2} - 374 T^{3} + 10642 T^{4} + 16154 T^{5} - 772597 T^{6} + 16154 p T^{7} + 10642 p^{2} T^{8} - 374 p^{3} T^{9} - 149 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 15 T - 51 T^{2} - 678 T^{3} + 17133 T^{4} + 80979 T^{5} - 925778 T^{6} + 80979 p T^{7} + 17133 p^{2} T^{8} - 678 p^{3} T^{9} - 51 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
89 \( ( 1 - 3 T + p T^{2} )^{6} \)
97 \( 1 - 18 T + 69 T^{2} - 214 T^{3} + 906 T^{4} + 163158 T^{5} - 2743251 T^{6} + 163158 p T^{7} + 906 p^{2} T^{8} - 214 p^{3} T^{9} + 69 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.46613653579566294475003308056, −7.22394298586076588066659082467, −6.87027369207621642519436798943, −6.79965125248867165690370437399, −6.70588699635044621079410971689, −6.23644424228524570504671216909, −6.21594770802415375525296845687, −5.79238938142523179117719186111, −5.73358348662922249710160424923, −5.46225681079787195220808514930, −5.42861301970256754403482849925, −5.35186945034390950931574169800, −4.96691338397767880240480542183, −4.60388041085956442777889062723, −4.26288910081781160075540917074, −4.22357116927821807749007395439, −3.63996235304626027350577831368, −3.40989333584331581688922123384, −3.31589077377514844905698331998, −2.89777760502808939558800778458, −2.88410088745122794489392423958, −2.50066657728374814554970133046, −1.88553473442488638822934478143, −1.84135831178399842354635387809, −0.875053762900632261800193946079, 0.875053762900632261800193946079, 1.84135831178399842354635387809, 1.88553473442488638822934478143, 2.50066657728374814554970133046, 2.88410088745122794489392423958, 2.89777760502808939558800778458, 3.31589077377514844905698331998, 3.40989333584331581688922123384, 3.63996235304626027350577831368, 4.22357116927821807749007395439, 4.26288910081781160075540917074, 4.60388041085956442777889062723, 4.96691338397767880240480542183, 5.35186945034390950931574169800, 5.42861301970256754403482849925, 5.46225681079787195220808514930, 5.73358348662922249710160424923, 5.79238938142523179117719186111, 6.21594770802415375525296845687, 6.23644424228524570504671216909, 6.70588699635044621079410971689, 6.79965125248867165690370437399, 6.87027369207621642519436798943, 7.22394298586076588066659082467, 7.46613653579566294475003308056

Graph of the $Z$-function along the critical line

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