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Properties

Label 12-4002e6-1.1-c1e6-0-0
Degree $12$
Conductor $4.108\times 10^{21}$
Sign $1$
Analytic cond. $1.06494\times 10^{9}$
Root an. cond. $5.65297$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·2-s + 6·3-s + 21·4-s + 3·5-s − 36·6-s + 2·7-s − 56·8-s + 21·9-s − 18·10-s + 7·11-s + 126·12-s + 3·13-s − 12·14-s + 18·15-s + 126·16-s + 8·17-s − 126·18-s − 4·19-s + 63·20-s + 12·21-s − 42·22-s + 6·23-s − 336·24-s − 12·25-s − 18·26-s + 56·27-s + 42·28-s + ⋯
L(s)  = 1  − 4.24·2-s + 3.46·3-s + 21/2·4-s + 1.34·5-s − 14.6·6-s + 0.755·7-s − 19.7·8-s + 7·9-s − 5.69·10-s + 2.11·11-s + 36.3·12-s + 0.832·13-s − 3.20·14-s + 4.64·15-s + 63/2·16-s + 1.94·17-s − 29.6·18-s − 0.917·19-s + 14.0·20-s + 2.61·21-s − 8.95·22-s + 1.25·23-s − 68.5·24-s − 2.39·25-s − 3.53·26-s + 10.7·27-s + 7.93·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 23^{6} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 23^{6} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 23^{6} \cdot 29^{6}\)
Sign: $1$
Analytic conductor: \(1.06494\times 10^{9}\)
Root analytic conductor: \(5.65297\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 23^{6} \cdot 29^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(36.16034911\)
\(L(\frac12)\) \(\approx\) \(36.16034911\)
\(L(\frac{3}{2})\) 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 \( ( 1 - T )^{6} \)
23 \( ( 1 - T )^{6} \)
29 \( ( 1 - T )^{6} \)
good5 \( ( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} )^{3} \)
7 \( 1 - 2 T + 19 T^{2} - 30 T^{3} + 219 T^{4} - 264 T^{5} + 1682 T^{6} - 264 p T^{7} + 219 p^{2} T^{8} - 30 p^{3} T^{9} + 19 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 7 T + 72 T^{2} - 347 T^{3} + 2031 T^{4} - 7264 T^{5} + 29952 T^{6} - 7264 p T^{7} + 2031 p^{2} T^{8} - 347 p^{3} T^{9} + 72 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 3 T + 54 T^{2} - 177 T^{3} + 1371 T^{4} - 4424 T^{5} + 21772 T^{6} - 4424 p T^{7} + 1371 p^{2} T^{8} - 177 p^{3} T^{9} + 54 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 8 T + 83 T^{2} - 518 T^{3} + 3155 T^{4} - 15498 T^{5} + 69090 T^{6} - 15498 p T^{7} + 3155 p^{2} T^{8} - 518 p^{3} T^{9} + 83 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 4 T + 67 T^{2} + 350 T^{3} + 2463 T^{4} + 11710 T^{5} + 59282 T^{6} + 11710 p T^{7} + 2463 p^{2} T^{8} + 350 p^{3} T^{9} + 67 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - T + 84 T^{2} + 107 T^{3} + 3407 T^{4} + 11204 T^{5} + 105304 T^{6} + 11204 p T^{7} + 3407 p^{2} T^{8} + 107 p^{3} T^{9} + 84 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 7 T + 145 T^{2} - 842 T^{3} + 10265 T^{4} - 51327 T^{5} + 461994 T^{6} - 51327 p T^{7} + 10265 p^{2} T^{8} - 842 p^{3} T^{9} + 145 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 13 T + 165 T^{2} - 1194 T^{3} + 9229 T^{4} - 55617 T^{5} + 387642 T^{6} - 55617 p T^{7} + 9229 p^{2} T^{8} - 1194 p^{3} T^{9} + 165 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 219 T^{2} + 50 T^{3} + 493 p T^{4} + 6070 T^{5} + 1172466 T^{6} + 6070 p T^{7} + 493 p^{3} T^{8} + 50 p^{3} T^{9} + 219 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 - 22 T + 443 T^{2} - 5526 T^{3} + 62811 T^{4} - 533680 T^{5} + 4153682 T^{6} - 533680 p T^{7} + 62811 p^{2} T^{8} - 5526 p^{3} T^{9} + 443 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 10 T + 230 T^{2} - 1882 T^{3} + 23799 T^{4} - 163908 T^{5} + 1527284 T^{6} - 163908 p T^{7} + 23799 p^{2} T^{8} - 1882 p^{3} T^{9} + 230 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 17 T + 245 T^{2} - 1866 T^{3} + 13171 T^{4} - 39341 T^{5} + 274958 T^{6} - 39341 p T^{7} + 13171 p^{2} T^{8} - 1866 p^{3} T^{9} + 245 p^{4} T^{10} - 17 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - T + 172 T^{2} - 153 T^{3} + 15631 T^{4} - 26114 T^{5} + 1085944 T^{6} - 26114 p T^{7} + 15631 p^{2} T^{8} - 153 p^{3} T^{9} + 172 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 3 T + 244 T^{2} - 697 T^{3} + 30199 T^{4} - 78874 T^{5} + 2432664 T^{6} - 78874 p T^{7} + 30199 p^{2} T^{8} - 697 p^{3} T^{9} + 244 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 11 T + 344 T^{2} - 2579 T^{3} + 48599 T^{4} - 277464 T^{5} + 4173216 T^{6} - 277464 p T^{7} + 48599 p^{2} T^{8} - 2579 p^{3} T^{9} + 344 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 282 T^{2} - 480 T^{3} + 35775 T^{4} - 105760 T^{5} + 2995564 T^{6} - 105760 p T^{7} + 35775 p^{2} T^{8} - 480 p^{3} T^{9} + 282 p^{4} T^{10} + p^{6} T^{12} \)
79 \( 1 - 2 T - 16 T^{2} - 138 T^{3} + 18991 T^{4} - 23464 T^{5} - 200704 T^{6} - 23464 p T^{7} + 18991 p^{2} T^{8} - 138 p^{3} T^{9} - 16 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 16 T + 518 T^{2} - 5724 T^{3} + 103567 T^{4} - 857596 T^{5} + 11183492 T^{6} - 857596 p T^{7} + 103567 p^{2} T^{8} - 5724 p^{3} T^{9} + 518 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 16 T + 446 T^{2} - 5376 T^{3} + 82239 T^{4} - 803760 T^{5} + 8999332 T^{6} - 803760 p T^{7} + 82239 p^{2} T^{8} - 5376 p^{3} T^{9} + 446 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 2 T + 290 T^{2} - 342 T^{3} + 49863 T^{4} - 57272 T^{5} + 5865804 T^{6} - 57272 p T^{7} + 49863 p^{2} T^{8} - 342 p^{3} T^{9} + 290 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.21056237963743952730465038818, −3.79363567087771109781251529602, −3.78071530237853460264202644882, −3.74777876793980576978771105976, −3.70981552321667188876355036032, −3.69979807030575537424708662067, −3.48455762967086227990975136526, −3.00662467895420760662769086181, −2.95001307629233913367159236299, −2.80978638299004530061618342195, −2.78366587588864850893281696329, −2.69583237779665962817186607422, −2.42266334594031928169210101362, −2.07737583959474543666899290444, −1.99818618069142297602772310706, −1.92491183332700487710984118619, −1.92386203276090569725609376749, −1.89331894051472742213117147580, −1.58900692655119686471271297851, −1.35714042078149791784825366246, −1.04303838184441728380202124826, −0.890616669427389115564061231080, −0.78386415290010982350728773879, −0.73263615379500269157525798365, −0.63298066853845904301300507442, 0.63298066853845904301300507442, 0.73263615379500269157525798365, 0.78386415290010982350728773879, 0.890616669427389115564061231080, 1.04303838184441728380202124826, 1.35714042078149791784825366246, 1.58900692655119686471271297851, 1.89331894051472742213117147580, 1.92386203276090569725609376749, 1.92491183332700487710984118619, 1.99818618069142297602772310706, 2.07737583959474543666899290444, 2.42266334594031928169210101362, 2.69583237779665962817186607422, 2.78366587588864850893281696329, 2.80978638299004530061618342195, 2.95001307629233913367159236299, 3.00662467895420760662769086181, 3.48455762967086227990975136526, 3.69979807030575537424708662067, 3.70981552321667188876355036032, 3.74777876793980576978771105976, 3.78071530237853460264202644882, 3.79363567087771109781251529602, 4.21056237963743952730465038818

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

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