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Properties

Label 12-1368e6-1.1-c1e6-0-1
Degree $12$
Conductor $6.554\times 10^{18}$
Sign $1$
Analytic cond. $1.69894\times 10^{6}$
Root an. cond. $3.30507$
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  + 6·7-s + 12·11-s − 3·13-s + 3·25-s + 12·29-s + 30·31-s − 6·37-s + 12·41-s − 9·43-s − 18·47-s + 3·49-s − 6·59-s + 3·61-s + 3·67-s + 6·71-s + 9·73-s + 72·77-s − 27·79-s + 24·83-s + 18·89-s − 18·91-s − 6·97-s + 6·101-s − 18·103-s − 12·107-s − 18·109-s + 12·113-s + ⋯
L(s)  = 1  + 2.26·7-s + 3.61·11-s − 0.832·13-s + 3/5·25-s + 2.22·29-s + 5.38·31-s − 0.986·37-s + 1.87·41-s − 1.37·43-s − 2.62·47-s + 3/7·49-s − 0.781·59-s + 0.384·61-s + 0.366·67-s + 0.712·71-s + 1.05·73-s + 8.20·77-s − 3.03·79-s + 2.63·83-s + 1.90·89-s − 1.88·91-s − 0.609·97-s + 0.597·101-s − 1.77·103-s − 1.16·107-s − 1.72·109-s + 1.12·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 3^{12} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(1.69894\times 10^{6}\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 3^{12} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(16.14506252\)
\(L(\frac12)\) \(\approx\) \(16.14506252\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + 9 T^{2} + 64 T^{3} + 9 p T^{4} + p^{3} T^{6} \)
good5 \( 1 - 3 T^{2} - 16 T^{3} - 6 T^{4} + 24 T^{5} + 269 T^{6} + 24 p T^{7} - 6 p^{2} T^{8} - 16 p^{3} T^{9} - 3 p^{4} T^{10} + p^{6} T^{12} \)
7 \( ( 1 - 3 T + 12 T^{2} - 39 T^{3} + 12 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 - 6 T + 9 T^{2} + 4 T^{3} + 9 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( ( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} )^{3} \)
17 \( 1 - 3 T^{2} - 128 T^{3} - 42 T^{4} + 192 T^{5} + 13769 T^{6} + 192 p T^{7} - 42 p^{2} T^{8} - 128 p^{3} T^{9} - 3 p^{4} T^{10} + p^{6} T^{12} \)
23 \( 1 - 57 T^{2} - 16 T^{3} + 1938 T^{4} + 456 T^{5} - 50329 T^{6} + 456 p T^{7} + 1938 p^{2} T^{8} - 16 p^{3} T^{9} - 57 p^{4} T^{10} + p^{6} T^{12} \)
29 \( 1 - 12 T + 57 T^{2} - 36 T^{3} - 1002 T^{4} + 228 p T^{5} - 37811 T^{6} + 228 p^{2} T^{7} - 1002 p^{2} T^{8} - 36 p^{3} T^{9} + 57 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 - 15 T + 132 T^{2} - 803 T^{3} + 132 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( ( 1 + 3 T + 66 T^{2} + 3 p T^{3} + 66 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 - 12 T + 21 T^{2} + 108 T^{3} + 582 T^{4} + 84 p T^{5} - 97247 T^{6} + 84 p^{2} T^{7} + 582 p^{2} T^{8} + 108 p^{3} T^{9} + 21 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 9 T - 63 T^{2} - 218 T^{3} + 7731 T^{4} + 14193 T^{5} - 309354 T^{6} + 14193 p T^{7} + 7731 p^{2} T^{8} - 218 p^{3} T^{9} - 63 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
47 \( ( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{3} \)
53 \( 1 - 75 T^{2} - 592 T^{3} + 1650 T^{4} + 22200 T^{5} + 87245 T^{6} + 22200 p T^{7} + 1650 p^{2} T^{8} - 592 p^{3} T^{9} - 75 p^{4} T^{10} + p^{6} T^{12} \)
59 \( 1 + 6 T - 45 T^{2} - 82 T^{3} + 78 T^{4} - 13458 T^{5} - 21001 T^{6} - 13458 p T^{7} + 78 p^{2} T^{8} - 82 p^{3} T^{9} - 45 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 3 T - 129 T^{2} + 352 T^{3} + 9477 T^{4} - 13509 T^{5} - 594522 T^{6} - 13509 p T^{7} + 9477 p^{2} T^{8} + 352 p^{3} T^{9} - 129 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 3 T - 159 T^{2} + 374 T^{3} + 15651 T^{4} - 19683 T^{5} - 1137162 T^{6} - 19683 p T^{7} + 15651 p^{2} T^{8} + 374 p^{3} T^{9} - 159 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 6 T - 45 T^{2} - 494 T^{3} + 834 T^{4} + 39090 T^{5} + 68531 T^{6} + 39090 p T^{7} + 834 p^{2} T^{8} - 494 p^{3} T^{9} - 45 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 9 T - 117 T^{2} + 484 T^{3} + 14553 T^{4} - 12123 T^{5} - 1231818 T^{6} - 12123 p T^{7} + 14553 p^{2} T^{8} + 484 p^{3} T^{9} - 117 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 27 T + 333 T^{2} + 2806 T^{3} + 19071 T^{4} + 40599 T^{5} - 404826 T^{6} + 40599 p T^{7} + 19071 p^{2} T^{8} + 2806 p^{3} T^{9} + 333 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 - 12 T + 153 T^{2} - 2056 T^{3} + 153 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 18 T - 15 T^{2} + 162 T^{3} + 28374 T^{4} - 118170 T^{5} - 1575011 T^{6} - 118170 p T^{7} + 28374 p^{2} T^{8} + 162 p^{3} T^{9} - 15 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 6 T - 123 T^{2} + 338 T^{3} + 8010 T^{4} - 79650 T^{5} - 870771 T^{6} - 79650 p T^{7} + 8010 p^{2} T^{8} + 338 p^{3} T^{9} - 123 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.98355793472549819679874330248, −4.90078291673612469805789538801, −4.54377455418055608054767599199, −4.53342190598774434080154771440, −4.50124359781183489364474120124, −4.47288433553394648561383533638, −4.35133154099774654818488091527, −3.73283721723274827514032231770, −3.71096428534882567228945134019, −3.66318181835504861490790906913, −3.58813435629921523393983516132, −3.36104643187903926829099184267, −2.89965496176559572892509945095, −2.72461051280171794247916858762, −2.65091461773479678851816683072, −2.64228846187652217405153783509, −2.40821147042629645041435822305, −1.92264641989444094445282057784, −1.76666267177002707174778107922, −1.48990395070094719059366673024, −1.42564037333114073648210165034, −1.28898180626818330580976499530, −1.02771656043443994500905779604, −0.795024086808873161380527557363, −0.46707141351337917191615402969, 0.46707141351337917191615402969, 0.795024086808873161380527557363, 1.02771656043443994500905779604, 1.28898180626818330580976499530, 1.42564037333114073648210165034, 1.48990395070094719059366673024, 1.76666267177002707174778107922, 1.92264641989444094445282057784, 2.40821147042629645041435822305, 2.64228846187652217405153783509, 2.65091461773479678851816683072, 2.72461051280171794247916858762, 2.89965496176559572892509945095, 3.36104643187903926829099184267, 3.58813435629921523393983516132, 3.66318181835504861490790906913, 3.71096428534882567228945134019, 3.73283721723274827514032231770, 4.35133154099774654818488091527, 4.47288433553394648561383533638, 4.50124359781183489364474120124, 4.53342190598774434080154771440, 4.54377455418055608054767599199, 4.90078291673612469805789538801, 4.98355793472549819679874330248

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

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