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Properties

Label 12-5635e6-1.1-c1e6-0-0
Degree $12$
Conductor $3.202\times 10^{22}$
Sign $1$
Analytic cond. $8.29900\times 10^{9}$
Root an. cond. $6.70788$
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  − 4·4-s − 6·5-s + 2·8-s − 10·9-s + 4·11-s + 8·13-s + 7·16-s − 2·19-s + 24·20-s − 6·23-s + 21·25-s + 2·27-s − 8·29-s + 8·31-s − 8·32-s + 40·36-s − 16·37-s − 12·40-s + 22·41-s + 8·43-s − 16·44-s + 60·45-s − 2·47-s − 32·52-s + 4·53-s − 24·55-s + 18·59-s + ⋯
L(s)  = 1  − 2·4-s − 2.68·5-s + 0.707·8-s − 3.33·9-s + 1.20·11-s + 2.21·13-s + 7/4·16-s − 0.458·19-s + 5.36·20-s − 1.25·23-s + 21/5·25-s + 0.384·27-s − 1.48·29-s + 1.43·31-s − 1.41·32-s + 20/3·36-s − 2.63·37-s − 1.89·40-s + 3.43·41-s + 1.21·43-s − 2.41·44-s + 8.94·45-s − 0.291·47-s − 4.43·52-s + 0.549·53-s − 3.23·55-s + 2.34·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 7^{12} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 7^{12} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(5^{6} \cdot 7^{12} \cdot 23^{6}\)
Sign: $1$
Analytic conductor: \(8.29900\times 10^{9}\)
Root analytic conductor: \(6.70788\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 5^{6} \cdot 7^{12} \cdot 23^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.165332149\)
\(L(\frac12)\) \(\approx\) \(1.165332149\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( ( 1 + T )^{6} \)
7 \( 1 \)
23 \( ( 1 + T )^{6} \)
good2 \( 1 + p^{2} T^{2} - p T^{3} + 9 T^{4} - p^{3} T^{5} + 19 T^{6} - p^{4} T^{7} + 9 p^{2} T^{8} - p^{4} T^{9} + p^{6} T^{10} + p^{6} T^{12} \)
3 \( 1 + 10 T^{2} - 2 T^{3} + 52 T^{4} - 14 T^{5} + 185 T^{6} - 14 p T^{7} + 52 p^{2} T^{8} - 2 p^{3} T^{9} + 10 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 4 T + 51 T^{2} - 186 T^{3} + 1217 T^{4} - 3764 T^{5} + 17067 T^{6} - 3764 p T^{7} + 1217 p^{2} T^{8} - 186 p^{3} T^{9} + 51 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 8 T + 83 T^{2} - 32 p T^{3} + 2493 T^{4} - 9206 T^{5} + 41109 T^{6} - 9206 p T^{7} + 2493 p^{2} T^{8} - 32 p^{4} T^{9} + 83 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 75 T^{2} - 16 T^{3} + 2671 T^{4} - 584 T^{5} + 57369 T^{6} - 584 p T^{7} + 2671 p^{2} T^{8} - 16 p^{3} T^{9} + 75 p^{4} T^{10} + p^{6} T^{12} \)
19 \( 1 + 2 T + 68 T^{2} + 124 T^{3} + 2380 T^{4} + 3904 T^{5} + 54299 T^{6} + 3904 p T^{7} + 2380 p^{2} T^{8} + 124 p^{3} T^{9} + 68 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 8 T + 90 T^{2} + 424 T^{3} + 3223 T^{4} + 9584 T^{5} + 77356 T^{6} + 9584 p T^{7} + 3223 p^{2} T^{8} + 424 p^{3} T^{9} + 90 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 8 T + 88 T^{2} - 530 T^{3} + 4548 T^{4} - 23208 T^{5} + 155815 T^{6} - 23208 p T^{7} + 4548 p^{2} T^{8} - 530 p^{3} T^{9} + 88 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 16 T + 207 T^{2} + 36 p T^{3} + 7159 T^{4} + 280 p T^{5} + 43641 T^{6} + 280 p^{2} T^{7} + 7159 p^{2} T^{8} + 36 p^{4} T^{9} + 207 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 22 T + 9 p T^{2} - 3986 T^{3} + 37330 T^{4} - 275598 T^{5} + 1921585 T^{6} - 275598 p T^{7} + 37330 p^{2} T^{8} - 3986 p^{3} T^{9} + 9 p^{5} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 8 T + 142 T^{2} - 668 T^{3} + 8728 T^{4} - 25344 T^{5} + 364613 T^{6} - 25344 p T^{7} + 8728 p^{2} T^{8} - 668 p^{3} T^{9} + 142 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 2 T + 108 T^{2} + 550 T^{3} + 7810 T^{4} + 42540 T^{5} + 422981 T^{6} + 42540 p T^{7} + 7810 p^{2} T^{8} + 550 p^{3} T^{9} + 108 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 4 T + 5 p T^{2} - 1040 T^{3} + 31107 T^{4} - 109216 T^{5} + 2106141 T^{6} - 109216 p T^{7} + 31107 p^{2} T^{8} - 1040 p^{3} T^{9} + 5 p^{5} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 18 T + 363 T^{2} - 4576 T^{3} + 53241 T^{4} - 498410 T^{5} + 4168111 T^{6} - 498410 p T^{7} + 53241 p^{2} T^{8} - 4576 p^{3} T^{9} + 363 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 6 T + 239 T^{2} + 1826 T^{3} + 25721 T^{4} + 221780 T^{5} + 1813291 T^{6} + 221780 p T^{7} + 25721 p^{2} T^{8} + 1826 p^{3} T^{9} + 239 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 4 T + 171 T^{2} - 1064 T^{3} + 18953 T^{4} - 104104 T^{5} + 1597637 T^{6} - 104104 p T^{7} + 18953 p^{2} T^{8} - 1064 p^{3} T^{9} + 171 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 240 T^{2} - 790 T^{3} + 26668 T^{4} - 149552 T^{5} + 2079531 T^{6} - 149552 p T^{7} + 26668 p^{2} T^{8} - 790 p^{3} T^{9} + 240 p^{4} T^{10} + p^{6} T^{12} \)
73 \( 1 - 12 T + 217 T^{2} - 1750 T^{3} + 23127 T^{4} - 175534 T^{5} + 2056377 T^{6} - 175534 p T^{7} + 23127 p^{2} T^{8} - 1750 p^{3} T^{9} + 217 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 30 T + 724 T^{2} + 11608 T^{3} + 163580 T^{4} + 1807810 T^{5} + 17885911 T^{6} + 1807810 p T^{7} + 163580 p^{2} T^{8} + 11608 p^{3} T^{9} + 724 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 32 T + 723 T^{2} - 12072 T^{3} + 163757 T^{4} - 22520 p T^{5} + 18340989 T^{6} - 22520 p^{2} T^{7} + 163757 p^{2} T^{8} - 12072 p^{3} T^{9} + 723 p^{4} T^{10} - 32 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 28 T + 711 T^{2} + 11810 T^{3} + 176957 T^{4} + 2055580 T^{5} + 21638875 T^{6} + 2055580 p T^{7} + 176957 p^{2} T^{8} + 11810 p^{3} T^{9} + 711 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 4 T + 3 p T^{2} - 660 T^{3} + 50746 T^{4} - 88116 T^{5} + 5811747 T^{6} - 88116 p T^{7} + 50746 p^{2} T^{8} - 660 p^{3} T^{9} + 3 p^{5} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.02876116833904805195079029487, −3.89103195599180565192052710325, −3.87945843565687107707802692607, −3.81546201311617002516210250339, −3.78577983355119517448078899320, −3.66592931571378004817506554628, −3.58581035964377312414920414484, −3.11183275912858740795974835030, −3.02536949421090340850698505290, −2.97342428910089706984913937878, −2.96134459903182314593907083863, −2.55858909958580967212453571792, −2.49019588452961118176957494371, −2.37631597258034886533149027668, −2.35086719832790921040311519421, −1.97260496667229147635144131907, −1.48356513704503859305587423947, −1.45318544071861225552849937799, −1.44027231068841034910979491605, −1.28987544652306117065595251917, −0.823332945651355994430361099734, −0.69409772328510624519236716210, −0.45547175936819650079389374588, −0.34164948876921422129170877397, −0.28479904257722047881709395135, 0.28479904257722047881709395135, 0.34164948876921422129170877397, 0.45547175936819650079389374588, 0.69409772328510624519236716210, 0.823332945651355994430361099734, 1.28987544652306117065595251917, 1.44027231068841034910979491605, 1.45318544071861225552849937799, 1.48356513704503859305587423947, 1.97260496667229147635144131907, 2.35086719832790921040311519421, 2.37631597258034886533149027668, 2.49019588452961118176957494371, 2.55858909958580967212453571792, 2.96134459903182314593907083863, 2.97342428910089706984913937878, 3.02536949421090340850698505290, 3.11183275912858740795974835030, 3.58581035964377312414920414484, 3.66592931571378004817506554628, 3.78577983355119517448078899320, 3.81546201311617002516210250339, 3.87945843565687107707802692607, 3.89103195599180565192052710325, 4.02876116833904805195079029487

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

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