↑

Properties

Label 12-4026e6-1.1-c1e6-0-0
Degree $12$
Conductor $4.258\times 10^{21}$
Sign $1$
Analytic cond. $1.10383\times 10^{9}$
Root an. cond. $5.66990$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 6·2-s − 6·3-s + 21·4-s − 5-s + 36·6-s + 5·7-s − 56·8-s + 21·9-s + 6·10-s + 6·11-s − 126·12-s + 2·13-s − 30·14-s + 6·15-s + 126·16-s − 4·17-s − 126·18-s − 5·19-s − 21·20-s − 30·21-s − 36·22-s − 6·23-s + 336·24-s − 15·25-s − 12·26-s − 56·27-s + 105·28-s + ⋯
L(s)  = 1  − 4.24·2-s − 3.46·3-s + 21/2·4-s − 0.447·5-s + 14.6·6-s + 1.88·7-s − 19.7·8-s + 7·9-s + 1.89·10-s + 1.80·11-s − 36.3·12-s + 0.554·13-s − 8.01·14-s + 1.54·15-s + 63/2·16-s − 0.970·17-s − 29.6·18-s − 1.14·19-s − 4.69·20-s − 6.54·21-s − 7.67·22-s − 1.25·23-s + 68.5·24-s − 3·25-s − 2.35·26-s − 10.7·27-s + 19.8·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 11^{6} \cdot 61^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 11^{6} \cdot 61^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 11^{6} \cdot 61^{6}\)
Sign: $1$
Analytic conductor: \(1.10383\times 10^{9}\)
Root analytic conductor: \(5.66990\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 11^{6} \cdot 61^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T )^{6} \)
3 \( ( 1 + T )^{6} \)
11 \( ( 1 - T )^{6} \)
61 \( ( 1 - T )^{6} \)
good5 \( 1 + T + 16 T^{2} + 22 T^{3} + 127 T^{4} + 213 T^{5} + 712 T^{6} + 213 p T^{7} + 127 p^{2} T^{8} + 22 p^{3} T^{9} + 16 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 5 T + 32 T^{2} - 93 T^{3} + 407 T^{4} - 1009 T^{5} + 3604 T^{6} - 1009 p T^{7} + 407 p^{2} T^{8} - 93 p^{3} T^{9} + 32 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + 462 T^{4} + 219 T^{5} + 4820 T^{6} + 219 p T^{7} + 462 p^{2} T^{8} - 2 p^{4} T^{9} + 34 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 4 T + 91 T^{2} + 279 T^{3} + 3523 T^{4} + 8442 T^{5} + 77012 T^{6} + 8442 p T^{7} + 3523 p^{2} T^{8} + 279 p^{3} T^{9} + 91 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 5 T + 60 T^{2} + 225 T^{3} + 1851 T^{4} + 5521 T^{5} + 37764 T^{6} + 5521 p T^{7} + 1851 p^{2} T^{8} + 225 p^{3} T^{9} + 60 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 6 T + 79 T^{2} + 341 T^{3} + 3195 T^{4} + 11719 T^{5} + 88522 T^{6} + 11719 p T^{7} + 3195 p^{2} T^{8} + 341 p^{3} T^{9} + 79 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 10 T + 153 T^{2} + 1149 T^{3} + 9903 T^{4} + 58129 T^{5} + 365798 T^{6} + 58129 p T^{7} + 9903 p^{2} T^{8} + 1149 p^{3} T^{9} + 153 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 15 T + 180 T^{2} + 1096 T^{3} + 5509 T^{4} + 9033 T^{5} + 36380 T^{6} + 9033 p T^{7} + 5509 p^{2} T^{8} + 1096 p^{3} T^{9} + 180 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 3 T + 76 T^{2} + 119 T^{3} + 4159 T^{4} + 5613 T^{5} + 166518 T^{6} + 5613 p T^{7} + 4159 p^{2} T^{8} + 119 p^{3} T^{9} + 76 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 9 T + 199 T^{2} + 1200 T^{3} + 16009 T^{4} + 73479 T^{5} + 787486 T^{6} + 73479 p T^{7} + 16009 p^{2} T^{8} + 1200 p^{3} T^{9} + 199 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 10 T + 199 T^{2} - 1345 T^{3} + 15837 T^{4} - 82771 T^{5} + 790822 T^{6} - 82771 p T^{7} + 15837 p^{2} T^{8} - 1345 p^{3} T^{9} + 199 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 14 T + 142 T^{2} + 948 T^{3} + 7946 T^{4} + 74593 T^{5} + 596918 T^{6} + 74593 p T^{7} + 7946 p^{2} T^{8} + 948 p^{3} T^{9} + 142 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 11 T + 306 T^{2} + 2587 T^{3} + 39037 T^{4} + 257183 T^{5} + 2716878 T^{6} + 257183 p T^{7} + 39037 p^{2} T^{8} + 2587 p^{3} T^{9} + 306 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 20 T + 388 T^{2} + 5090 T^{3} + 59534 T^{4} + 9457 p T^{5} + 4732406 T^{6} + 9457 p^{2} T^{7} + 59534 p^{2} T^{8} + 5090 p^{3} T^{9} + 388 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 14 T + 298 T^{2} - 2778 T^{3} + 37102 T^{4} - 274023 T^{5} + 2934510 T^{6} - 274023 p T^{7} + 37102 p^{2} T^{8} - 2778 p^{3} T^{9} + 298 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 21 T + 456 T^{2} + 6236 T^{3} + 82897 T^{4} + 822655 T^{5} + 7859980 T^{6} + 822655 p T^{7} + 82897 p^{2} T^{8} + 6236 p^{3} T^{9} + 456 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 16 T + 391 T^{2} - 5055 T^{3} + 69949 T^{4} - 676281 T^{5} + 6823446 T^{6} - 676281 p T^{7} + 69949 p^{2} T^{8} - 5055 p^{3} T^{9} + 391 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 6 T + 162 T^{2} + 484 T^{3} + 16196 T^{4} + 36935 T^{5} + 1488670 T^{6} + 36935 p T^{7} + 16196 p^{2} T^{8} + 484 p^{3} T^{9} + 162 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 10 T + 355 T^{2} + 2481 T^{3} + 55667 T^{4} + 305639 T^{5} + 5550770 T^{6} + 305639 p T^{7} + 55667 p^{2} T^{8} + 2481 p^{3} T^{9} + 355 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 11 T + 348 T^{2} + 3384 T^{3} + 51315 T^{4} + 477593 T^{5} + 5047832 T^{6} + 477593 p T^{7} + 51315 p^{2} T^{8} + 3384 p^{3} T^{9} + 348 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 2 T + 334 T^{2} - 368 T^{3} + 51532 T^{4} - 38707 T^{5} + 5558388 T^{6} - 38707 p T^{7} + 51532 p^{2} T^{8} - 368 p^{3} T^{9} + 334 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.85673290011215092494825493424, −4.54119660902399518806844073757, −4.46168188376189661945895024576, −4.31067021768206019319715827589, −4.19524504464483957966184130907, −4.17249150865312099356696285163, −4.16297130642549108238172233413, −3.60558767344153025746327592912, −3.52638932553971656892664249128, −3.38721095397078897541326268346, −3.35975057387512319160515525061, −3.32415692565505673317790548098, −3.21136820300652684212357880163, −2.37257620465146469871943569868, −2.22209271726599498837237613990, −2.19552113860245821206072147239, −2.08875453244963999341472415709, −2.08000419921983085909072705916, −2.06164566843275293820219087746, −1.48536966765541131427338374957, −1.43851820951758176166911953819, −1.33353274232697063375701490253, −1.23882970412425720108910095206, −1.14267799313972869119861296923, −1.10726875054537707796267785005, 0, 0, 0, 0, 0, 0, 1.10726875054537707796267785005, 1.14267799313972869119861296923, 1.23882970412425720108910095206, 1.33353274232697063375701490253, 1.43851820951758176166911953819, 1.48536966765541131427338374957, 2.06164566843275293820219087746, 2.08000419921983085909072705916, 2.08875453244963999341472415709, 2.19552113860245821206072147239, 2.22209271726599498837237613990, 2.37257620465146469871943569868, 3.21136820300652684212357880163, 3.32415692565505673317790548098, 3.35975057387512319160515525061, 3.38721095397078897541326268346, 3.52638932553971656892664249128, 3.60558767344153025746327592912, 4.16297130642549108238172233413, 4.17249150865312099356696285163, 4.19524504464483957966184130907, 4.31067021768206019319715827589, 4.46168188376189661945895024576, 4.54119660902399518806844073757, 4.85673290011215092494825493424

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

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