Properties

Label 12-6664e6-1.1-c1e6-0-4
Degree $12$
Conductor $8.758\times 10^{22}$
Sign $1$
Analytic cond. $2.27024\times 10^{10}$
Root an. cond. $7.29467$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 8·9-s − 6·11-s − 2·13-s + 15-s − 6·17-s + 2·19-s − 5·23-s − 14·25-s − 12·27-s − 15·29-s + 8·31-s − 6·33-s − 16·37-s − 2·39-s + 4·41-s + 24·43-s − 8·45-s − 47-s − 6·51-s − 32·53-s − 6·55-s + 2·57-s − 6·59-s + 10·61-s − 2·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 8/3·9-s − 1.80·11-s − 0.554·13-s + 0.258·15-s − 1.45·17-s + 0.458·19-s − 1.04·23-s − 2.79·25-s − 2.30·27-s − 2.78·29-s + 1.43·31-s − 1.04·33-s − 2.63·37-s − 0.320·39-s + 0.624·41-s + 3.65·43-s − 1.19·45-s − 0.145·47-s − 0.840·51-s − 4.39·53-s − 0.809·55-s + 0.264·57-s − 0.781·59-s + 1.28·61-s − 0.248·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{12} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{12} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 7^{12} \cdot 17^{6}\)
Sign: $1$
Analytic conductor: \(2.27024\times 10^{10}\)
Root analytic conductor: \(7.29467\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{18} \cdot 7^{12} \cdot 17^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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 \( 1 \)
7 \( 1 \)
17 \( ( 1 + T )^{6} \)
good3 \( 1 - T + p^{2} T^{2} - 5 T^{3} + 5 p^{2} T^{4} - 23 T^{5} + 164 T^{6} - 23 p T^{7} + 5 p^{4} T^{8} - 5 p^{3} T^{9} + p^{6} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - T + 3 p T^{2} - 13 T^{3} + 112 T^{4} - 106 T^{5} + 624 T^{6} - 106 p T^{7} + 112 p^{2} T^{8} - 13 p^{3} T^{9} + 3 p^{5} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 6 T + 56 T^{2} + 235 T^{3} + 1306 T^{4} + 4282 T^{5} + 17906 T^{6} + 4282 p T^{7} + 1306 p^{2} T^{8} + 235 p^{3} T^{9} + 56 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 2 T + 4 p T^{2} + 83 T^{3} + 1336 T^{4} + 1862 T^{5} + 21608 T^{6} + 1862 p T^{7} + 1336 p^{2} T^{8} + 83 p^{3} T^{9} + 4 p^{5} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 2 T + 54 T^{2} - 144 T^{3} + 1422 T^{4} - 4914 T^{5} + 28114 T^{6} - 4914 p T^{7} + 1422 p^{2} T^{8} - 144 p^{3} T^{9} + 54 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 5 T + 120 T^{2} + 515 T^{3} + 6391 T^{4} + 970 p T^{5} + 191328 T^{6} + 970 p^{2} T^{7} + 6391 p^{2} T^{8} + 515 p^{3} T^{9} + 120 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 15 T + 219 T^{2} + 1988 T^{3} + 16973 T^{4} + 109573 T^{5} + 665822 T^{6} + 109573 p T^{7} + 16973 p^{2} T^{8} + 1988 p^{3} T^{9} + 219 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 8 T + 85 T^{2} - 363 T^{3} + 3087 T^{4} - 16099 T^{5} + 125542 T^{6} - 16099 p T^{7} + 3087 p^{2} T^{8} - 363 p^{3} T^{9} + 85 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 16 T + 241 T^{2} + 2437 T^{3} + 22234 T^{4} + 163483 T^{5} + 1085740 T^{6} + 163483 p T^{7} + 22234 p^{2} T^{8} + 2437 p^{3} T^{9} + 241 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 4 T + 141 T^{2} + 13 T^{3} + 6809 T^{4} + 30719 T^{5} + 232690 T^{6} + 30719 p T^{7} + 6809 p^{2} T^{8} + 13 p^{3} T^{9} + 141 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 24 T + 392 T^{2} - 4744 T^{3} + 47200 T^{4} - 389136 T^{5} + 2769910 T^{6} - 389136 p T^{7} + 47200 p^{2} T^{8} - 4744 p^{3} T^{9} + 392 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + T + 121 T^{2} - 274 T^{3} + 5971 T^{4} - 47467 T^{5} + 223270 T^{6} - 47467 p T^{7} + 5971 p^{2} T^{8} - 274 p^{3} T^{9} + 121 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 32 T + 628 T^{2} + 8871 T^{3} + 100184 T^{4} + 938804 T^{5} + 7419360 T^{6} + 938804 p T^{7} + 100184 p^{2} T^{8} + 8871 p^{3} T^{9} + 628 p^{4} T^{10} + 32 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 6 T + 245 T^{2} + 1219 T^{3} + 29004 T^{4} + 125565 T^{5} + 36236 p T^{6} + 125565 p T^{7} + 29004 p^{2} T^{8} + 1219 p^{3} T^{9} + 245 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 10 T + 177 T^{2} - 1595 T^{3} + 20965 T^{4} - 151147 T^{5} + 1511354 T^{6} - 151147 p T^{7} + 20965 p^{2} T^{8} - 1595 p^{3} T^{9} + 177 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 27 T + 553 T^{2} - 7473 T^{3} + 89458 T^{4} - 856550 T^{5} + 7680708 T^{6} - 856550 p T^{7} + 89458 p^{2} T^{8} - 7473 p^{3} T^{9} + 553 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 12 T + 401 T^{2} + 3752 T^{3} + 68542 T^{4} + 501020 T^{5} + 6402797 T^{6} + 501020 p T^{7} + 68542 p^{2} T^{8} + 3752 p^{3} T^{9} + 401 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 15 T + 358 T^{2} - 4116 T^{3} + 58789 T^{4} - 525485 T^{5} + 5560680 T^{6} - 525485 p T^{7} + 58789 p^{2} T^{8} - 4116 p^{3} T^{9} + 358 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 21 T + 555 T^{2} + 7479 T^{3} + 113673 T^{4} + 1117635 T^{5} + 12053518 T^{6} + 1117635 p T^{7} + 113673 p^{2} T^{8} + 7479 p^{3} T^{9} + 555 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 20 T + 489 T^{2} + 6117 T^{3} + 91479 T^{4} + 879945 T^{5} + 9837342 T^{6} + 879945 p T^{7} + 91479 p^{2} T^{8} + 6117 p^{3} T^{9} + 489 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 33 T + 861 T^{2} + 15588 T^{3} + 235931 T^{4} + 2855079 T^{5} + 29765630 T^{6} + 2855079 p T^{7} + 235931 p^{2} T^{8} + 15588 p^{3} T^{9} + 861 p^{4} T^{10} + 33 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 9 T + 430 T^{2} + 2940 T^{3} + 82765 T^{4} + 454215 T^{5} + 9821344 T^{6} + 454215 p T^{7} + 82765 p^{2} T^{8} + 2940 p^{3} T^{9} + 430 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.40575511776057012267532303888, −4.31878683694941275467850079585, −4.14139893261055475942344998007, −3.94383998119627602046165313101, −3.85266391878274018026939905146, −3.84346966853185316263829515663, −3.70902407694428732592793088806, −3.48456228295923674773917381937, −3.33978020016638698666352299850, −3.28905370081314690797532345499, −3.00843793426527520865929864452, −2.95219300692453178298894419696, −2.81379888816436962780655655024, −2.49383471755119497091373051077, −2.44409693399062889598354321935, −2.32992005755821467984228396744, −2.32512117035693073132237664348, −2.27694866981531645393266464282, −2.15460665513394937673359316220, −1.75359430616853665373723268758, −1.59993381688025629468283959224, −1.58126280198489359130236592681, −1.13301010339543162969432720222, −1.07095149512365733598647470662, −1.00064144077974133175013322389, 0, 0, 0, 0, 0, 0, 1.00064144077974133175013322389, 1.07095149512365733598647470662, 1.13301010339543162969432720222, 1.58126280198489359130236592681, 1.59993381688025629468283959224, 1.75359430616853665373723268758, 2.15460665513394937673359316220, 2.27694866981531645393266464282, 2.32512117035693073132237664348, 2.32992005755821467984228396744, 2.44409693399062889598354321935, 2.49383471755119497091373051077, 2.81379888816436962780655655024, 2.95219300692453178298894419696, 3.00843793426527520865929864452, 3.28905370081314690797532345499, 3.33978020016638698666352299850, 3.48456228295923674773917381937, 3.70902407694428732592793088806, 3.84346966853185316263829515663, 3.85266391878274018026939905146, 3.94383998119627602046165313101, 4.14139893261055475942344998007, 4.31878683694941275467850079585, 4.40575511776057012267532303888

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

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