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Properties

Label 12-3344e6-1.1-c1e6-0-0
Degree $12$
Conductor $1.398\times 10^{21}$
Sign $1$
Analytic cond. $3.62460\times 10^{8}$
Root an. cond. $5.16739$
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  + 4·3-s − 3·5-s + 3·7-s + 2·9-s + 6·11-s − 3·13-s − 12·15-s − 17-s − 6·19-s + 12·21-s + 4·23-s − 8·25-s − 12·27-s − 7·29-s + 31-s + 24·33-s − 9·35-s + 5·37-s − 12·39-s − 6·41-s + 16·43-s − 6·45-s + 4·47-s − 20·49-s − 4·51-s − 11·53-s − 18·55-s + ⋯
L(s)  = 1  + 2.30·3-s − 1.34·5-s + 1.13·7-s + 2/3·9-s + 1.80·11-s − 0.832·13-s − 3.09·15-s − 0.242·17-s − 1.37·19-s + 2.61·21-s + 0.834·23-s − 8/5·25-s − 2.30·27-s − 1.29·29-s + 0.179·31-s + 4.17·33-s − 1.52·35-s + 0.821·37-s − 1.92·39-s − 0.937·41-s + 2.43·43-s − 0.894·45-s + 0.583·47-s − 2.85·49-s − 0.560·51-s − 1.51·53-s − 2.42·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 11^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(3.62460\times 10^{8}\)
Root analytic conductor: \(5.16739\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 11^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.411262979\)
\(L(\frac12)\) \(\approx\) \(5.411262979\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( ( 1 - T )^{6} \)
19 \( ( 1 + T )^{6} \)
good3 \( 1 - 4 T + 14 T^{2} - 4 p^{2} T^{3} + 89 T^{4} - 20 p^{2} T^{5} + 337 T^{6} - 20 p^{3} T^{7} + 89 p^{2} T^{8} - 4 p^{5} T^{9} + 14 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 + 3 T + 17 T^{2} + 52 T^{3} + 156 T^{4} + 394 T^{5} + 958 T^{6} + 394 p T^{7} + 156 p^{2} T^{8} + 52 p^{3} T^{9} + 17 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 3 T + 29 T^{2} - 66 T^{3} + 388 T^{4} - 752 T^{5} + 3334 T^{6} - 752 p T^{7} + 388 p^{2} T^{8} - 66 p^{3} T^{9} + 29 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 3 T + 33 T^{2} + 80 T^{3} + 466 T^{4} + 758 T^{5} + 5144 T^{6} + 758 p T^{7} + 466 p^{2} T^{8} + 80 p^{3} T^{9} + 33 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + T + 20 T^{2} - 3 p T^{3} + 519 T^{4} - 190 T^{5} + 12680 T^{6} - 190 p T^{7} + 519 p^{2} T^{8} - 3 p^{4} T^{9} + 20 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 4 T + 61 T^{2} - 112 T^{3} + 1479 T^{4} + 996 T^{5} + 25734 T^{6} + 996 p T^{7} + 1479 p^{2} T^{8} - 112 p^{3} T^{9} + 61 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 7 T + 147 T^{2} + 882 T^{3} + 9726 T^{4} + 47840 T^{5} + 364788 T^{6} + 47840 p T^{7} + 9726 p^{2} T^{8} + 882 p^{3} T^{9} + 147 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - T + 115 T^{2} + 122 T^{3} + 5608 T^{4} + 15564 T^{5} + 186246 T^{6} + 15564 p T^{7} + 5608 p^{2} T^{8} + 122 p^{3} T^{9} + 115 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 5 T + 92 T^{2} - 493 T^{3} + 4383 T^{4} - 34498 T^{5} + 187608 T^{6} - 34498 p T^{7} + 4383 p^{2} T^{8} - 493 p^{3} T^{9} + 92 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 6 T + 61 T^{2} + 304 T^{3} + 3850 T^{4} + 11860 T^{5} + 97028 T^{6} + 11860 p T^{7} + 3850 p^{2} T^{8} + 304 p^{3} T^{9} + 61 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 16 T + 269 T^{2} - 2884 T^{3} + 29320 T^{4} - 228602 T^{5} + 1685620 T^{6} - 228602 p T^{7} + 29320 p^{2} T^{8} - 2884 p^{3} T^{9} + 269 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 4 T + 198 T^{2} - 604 T^{3} + 19023 T^{4} - 46232 T^{5} + 1112852 T^{6} - 46232 p T^{7} + 19023 p^{2} T^{8} - 604 p^{3} T^{9} + 198 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 11 T + 228 T^{2} + 2059 T^{3} + 25775 T^{4} + 187654 T^{5} + 1708264 T^{6} + 187654 p T^{7} + 25775 p^{2} T^{8} + 2059 p^{3} T^{9} + 228 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 18 T + 287 T^{2} - 3070 T^{3} + 30343 T^{4} - 248948 T^{5} + 1986402 T^{6} - 248948 p T^{7} + 30343 p^{2} T^{8} - 3070 p^{3} T^{9} + 287 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 16 T + 414 T^{2} - 4640 T^{3} + 66423 T^{4} - 550992 T^{5} + 5477604 T^{6} - 550992 p T^{7} + 66423 p^{2} T^{8} - 4640 p^{3} T^{9} + 414 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 18 T + 426 T^{2} - 5040 T^{3} + 69861 T^{4} - 614124 T^{5} + 6152163 T^{6} - 614124 p T^{7} + 69861 p^{2} T^{8} - 5040 p^{3} T^{9} + 426 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 7 T + 285 T^{2} - 1236 T^{3} + 34280 T^{4} - 89346 T^{5} + 2710178 T^{6} - 89346 p T^{7} + 34280 p^{2} T^{8} - 1236 p^{3} T^{9} + 285 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 21 T + 426 T^{2} + 5509 T^{3} + 69103 T^{4} + 673534 T^{5} + 6426092 T^{6} + 673534 p T^{7} + 69103 p^{2} T^{8} + 5509 p^{3} T^{9} + 426 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 22 T + 462 T^{2} - 6906 T^{3} + 86543 T^{4} - 949988 T^{5} + 8893444 T^{6} - 949988 p T^{7} + 86543 p^{2} T^{8} - 6906 p^{3} T^{9} + 462 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 16 T + 373 T^{2} - 4116 T^{3} + 56544 T^{4} - 512170 T^{5} + 5513020 T^{6} - 512170 p T^{7} + 56544 p^{2} T^{8} - 4116 p^{3} T^{9} + 373 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 13 T + 390 T^{2} + 4233 T^{3} + 69727 T^{4} + 647346 T^{5} + 7681876 T^{6} + 647346 p T^{7} + 69727 p^{2} T^{8} + 4233 p^{3} T^{9} + 390 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 15 T + 334 T^{2} + 2447 T^{3} + 36831 T^{4} + 148370 T^{5} + 3109412 T^{6} + 148370 p T^{7} + 36831 p^{2} T^{8} + 2447 p^{3} T^{9} + 334 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.46006045629973264748530640802, −4.28372177221619046001433567289, −3.95043851663601243894290011628, −3.92719954471507299255435539368, −3.81363550021013917763536756538, −3.73507815209854226300489929216, −3.72734332934737422427595096204, −3.28722071276850797779382956411, −3.27392169781421542762380876761, −3.19353872488499630600546846899, −2.91096512659384949506533354962, −2.85711174793307199484108071881, −2.82008542631433658807943530586, −2.39252976129864772480342280953, −2.19637289188845690605459435022, −2.02708795273781507568250668110, −2.01236809741968421995085304345, −1.98297966392096619703219094295, −1.92957895989215285813473565454, −1.43190378759237055537892297456, −1.30701342155216906148581503521, −0.833165468390809403412805853022, −0.66019590857252220916434770876, −0.62095505696641869837863423990, −0.18199942745580164886373080749, 0.18199942745580164886373080749, 0.62095505696641869837863423990, 0.66019590857252220916434770876, 0.833165468390809403412805853022, 1.30701342155216906148581503521, 1.43190378759237055537892297456, 1.92957895989215285813473565454, 1.98297966392096619703219094295, 2.01236809741968421995085304345, 2.02708795273781507568250668110, 2.19637289188845690605459435022, 2.39252976129864772480342280953, 2.82008542631433658807943530586, 2.85711174793307199484108071881, 2.91096512659384949506533354962, 3.19353872488499630600546846899, 3.27392169781421542762380876761, 3.28722071276850797779382956411, 3.72734332934737422427595096204, 3.73507815209854226300489929216, 3.81363550021013917763536756538, 3.92719954471507299255435539368, 3.95043851663601243894290011628, 4.28372177221619046001433567289, 4.46006045629973264748530640802

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

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