Properties

Degree $12$
Conductor $5.362\times 10^{18}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2·4-s − 10·5-s + 8-s + 10·10-s + 4·11-s + 3·13-s + 12·17-s − 3·19-s − 20·20-s − 4·22-s + 41·25-s − 3·26-s + 29-s − 3·31-s + 4·32-s − 12·34-s + 3·37-s + 3·38-s − 10·40-s + 22·41-s + 3·43-s + 8·44-s + 9·47-s − 41·50-s + 6·52-s − 18·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 4-s − 4.47·5-s + 0.353·8-s + 3.16·10-s + 1.20·11-s + 0.832·13-s + 2.91·17-s − 0.688·19-s − 4.47·20-s − 0.852·22-s + 41/5·25-s − 0.588·26-s + 0.185·29-s − 0.538·31-s + 0.707·32-s − 2.05·34-s + 0.493·37-s + 0.486·38-s − 1.58·40-s + 3.43·41-s + 0.457·43-s + 1.20·44-s + 1.31·47-s − 5.79·50-s + 0.832·52-s − 2.47·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{18} \cdot 7^{12}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1323} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{18} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.718310108\)
\(L(\frac12)\) \(\approx\) \(2.718310108\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + T - T^{2} - p^{2} T^{3} - 3 T^{4} + p T^{5} + 13 T^{6} + p^{2} T^{7} - 3 p^{2} T^{8} - p^{5} T^{9} - p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
5 \( ( 1 + p T + 17 T^{2} + 39 T^{3} + 17 p T^{4} + p^{3} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 - 2 T + 14 T^{2} + 3 T^{3} + 14 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )^{3} \)
17 \( 1 - 12 T + 54 T^{2} - 210 T^{3} + 1350 T^{4} - 5898 T^{5} + 19735 T^{6} - 5898 p T^{7} + 1350 p^{2} T^{8} - 210 p^{3} T^{9} + 54 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 3 T - 42 T^{2} - 61 T^{3} + 69 p T^{4} + 726 T^{5} - 27501 T^{6} + 726 p T^{7} + 69 p^{3} T^{8} - 61 p^{3} T^{9} - 42 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( ( 1 + 36 T^{2} + 9 T^{3} + 36 p T^{4} + p^{3} T^{6} )^{2} \)
29 \( 1 - T - 82 T^{2} + 31 T^{3} + 4425 T^{4} - 758 T^{5} - 148595 T^{6} - 758 p T^{7} + 4425 p^{2} T^{8} + 31 p^{3} T^{9} - 82 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 3 T - 60 T^{2} - 219 T^{3} + 1983 T^{4} + 4746 T^{5} - 51289 T^{6} + 4746 p T^{7} + 1983 p^{2} T^{8} - 219 p^{3} T^{9} - 60 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 3 T - 48 T^{2} + 435 T^{3} + 231 T^{4} - 8724 T^{5} + 60581 T^{6} - 8724 p T^{7} + 231 p^{2} T^{8} + 435 p^{3} T^{9} - 48 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 22 T + 206 T^{2} - 1802 T^{3} + 18432 T^{4} - 135116 T^{5} + 808243 T^{6} - 135116 p T^{7} + 18432 p^{2} T^{8} - 1802 p^{3} T^{9} + 206 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 3 T - 54 T^{2} + 569 T^{3} + 123 T^{4} - 13170 T^{5} + 115347 T^{6} - 13170 p T^{7} + 123 p^{2} T^{8} + 569 p^{3} T^{9} - 54 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 9 T - 6 T^{2} + 531 T^{3} - 2433 T^{4} - 3438 T^{5} + 104623 T^{6} - 3438 p T^{7} - 2433 p^{2} T^{8} + 531 p^{3} T^{9} - 6 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 18 T + 90 T^{2} + 378 T^{3} + 7848 T^{4} + 52668 T^{5} + 160459 T^{6} + 52668 p T^{7} + 7848 p^{2} T^{8} + 378 p^{3} T^{9} + 90 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 9 T - 90 T^{2} + 459 T^{3} + 10161 T^{4} - 20556 T^{5} - 598421 T^{6} - 20556 p T^{7} + 10161 p^{2} T^{8} + 459 p^{3} T^{9} - 90 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 6 T - 126 T^{2} - 358 T^{3} + 12372 T^{4} + 11472 T^{5} - 838653 T^{6} + 11472 p T^{7} + 12372 p^{2} T^{8} - 358 p^{3} T^{9} - 126 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 6 T^{2} + 1366 T^{3} + 438 T^{4} + 4098 T^{5} + 1065603 T^{6} + 4098 p T^{7} + 438 p^{2} T^{8} + 1366 p^{3} T^{9} + 6 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 + 9 T + 207 T^{2} + 1197 T^{3} + 207 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 3 T - 42 T^{2} + 1209 T^{3} - 3165 T^{4} - 28380 T^{5} + 1003961 T^{6} - 28380 p T^{7} - 3165 p^{2} T^{8} + 1209 p^{3} T^{9} - 42 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 15 T + 36 T^{2} - 367 T^{3} - 3225 T^{4} - 51726 T^{5} - 676905 T^{6} - 51726 p T^{7} - 3225 p^{2} T^{8} - 367 p^{3} T^{9} + 36 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 12 T - 144 T^{2} + 582 T^{3} + 34812 T^{4} - 90444 T^{5} - 2656433 T^{6} - 90444 p T^{7} + 34812 p^{2} T^{8} + 582 p^{3} T^{9} - 144 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 2 T - 112 T^{2} + 1238 T^{3} + 1662 T^{4} - 59806 T^{5} + 720895 T^{6} - 59806 p T^{7} + 1662 p^{2} T^{8} + 1238 p^{3} T^{9} - 112 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 3 T - 168 T^{2} - 573 T^{3} + 14223 T^{4} + 78504 T^{5} - 1297807 T^{6} + 78504 p T^{7} + 14223 p^{2} T^{8} - 573 p^{3} T^{9} - 168 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.83170432643784022425888152312, −4.67038054700404674091273163295, −4.60550802267338605211265872712, −4.58764418524973941029405561402, −4.41942334881816831144023843546, −4.18749640783575758825557304325, −4.05374605374222600874212608289, −3.88988073287463422834688528533, −3.85624823435101183952641076334, −3.47961128614014436624210476316, −3.46229032008104564983169046681, −3.38701446524073121807416545304, −3.27842358194788625540565651967, −3.01022195203214727651648329254, −2.76401802381129087944607339259, −2.70334208213133594237498182370, −2.13297887142701466242833420487, −1.99415505762733069542259324745, −1.98377374991436671372807473993, −1.46462478634798521006361985841, −1.29948187193346348284707298411, −1.13028062862217883309692968100, −0.64197793722663956859630046809, −0.58564802765911944437048301100, −0.48234140602606020994439297457, 0.48234140602606020994439297457, 0.58564802765911944437048301100, 0.64197793722663956859630046809, 1.13028062862217883309692968100, 1.29948187193346348284707298411, 1.46462478634798521006361985841, 1.98377374991436671372807473993, 1.99415505762733069542259324745, 2.13297887142701466242833420487, 2.70334208213133594237498182370, 2.76401802381129087944607339259, 3.01022195203214727651648329254, 3.27842358194788625540565651967, 3.38701446524073121807416545304, 3.46229032008104564983169046681, 3.47961128614014436624210476316, 3.85624823435101183952641076334, 3.88988073287463422834688528533, 4.05374605374222600874212608289, 4.18749640783575758825557304325, 4.41942334881816831144023843546, 4.58764418524973941029405561402, 4.60550802267338605211265872712, 4.67038054700404674091273163295, 4.83170432643784022425888152312

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

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