Properties

Degree $12$
Conductor $1.049\times 10^{18}$
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  + 2·3-s − 5-s − 3·7-s + 6·9-s + 2·11-s − 3·13-s − 2·15-s + 4·17-s − 6·19-s − 6·21-s + 14·23-s + 11·25-s + 7·27-s − 29-s − 3·31-s + 4·33-s + 3·35-s − 6·37-s − 6·39-s + 3·43-s − 6·45-s + 21·47-s + 3·49-s + 8·51-s + 12·53-s − 2·55-s − 12·57-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s − 1.13·7-s + 2·9-s + 0.603·11-s − 0.832·13-s − 0.516·15-s + 0.970·17-s − 1.37·19-s − 1.30·21-s + 2.91·23-s + 11/5·25-s + 1.34·27-s − 0.185·29-s − 0.538·31-s + 0.696·33-s + 0.507·35-s − 0.986·37-s − 0.960·39-s + 0.457·43-s − 0.894·45-s + 3.06·47-s + 3/7·49-s + 1.12·51-s + 1.64·53-s − 0.269·55-s − 1.58·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{12} \cdot 7^{6}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1008} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{12} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.765707005\)
\(L(\frac12)\) \(\approx\) \(7.765707005\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T - 2 T^{2} + p^{2} T^{3} - 2 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
7 \( ( 1 + T + T^{2} )^{3} \)
good5 \( 1 + T - 2 p T^{2} - 7 T^{3} + 57 T^{4} + 14 T^{5} - 299 T^{6} + 14 p T^{7} + 57 p^{2} T^{8} - 7 p^{3} T^{9} - 2 p^{5} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 2 T - 4 T^{2} - 46 T^{3} + 6 T^{4} + 230 T^{5} + 1699 T^{6} + 230 p T^{7} + 6 p^{2} T^{8} - 46 p^{3} T^{9} - 4 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 3 T + 3 T^{2} - 84 T^{3} - 15 p T^{4} + 345 T^{5} + 5006 T^{6} + 345 p T^{7} - 15 p^{3} T^{8} - 84 p^{3} T^{9} + 3 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( ( 1 - 2 T + 32 T^{2} - 21 T^{3} + 32 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( ( 1 + 3 T + 33 T^{2} + 35 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 14 T + 74 T^{2} - 358 T^{3} + 2628 T^{4} - 11188 T^{5} + 33943 T^{6} - 11188 p T^{7} + 2628 p^{2} T^{8} - 358 p^{3} T^{9} + 74 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + T - 46 T^{2} + 149 T^{3} + 897 T^{4} - 4282 T^{5} - 13523 T^{6} - 4282 p T^{7} + 897 p^{2} T^{8} + 149 p^{3} T^{9} - 46 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 3 T - 48 T^{2} - 147 T^{3} + 1005 T^{4} + 1344 T^{5} - 24505 T^{6} + 1344 p T^{7} + 1005 p^{2} T^{8} - 147 p^{3} T^{9} - 48 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 3 T + 81 T^{2} + 199 T^{3} + 81 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 - 90 T^{2} + 18 T^{3} + 4410 T^{4} - 810 T^{5} - 194177 T^{6} - 810 p T^{7} + 4410 p^{2} T^{8} + 18 p^{3} T^{9} - 90 p^{4} T^{10} + p^{6} T^{12} \)
43 \( 1 - 3 T - 24 T^{2} + 979 T^{3} - 1947 T^{4} - 14820 T^{5} + 386067 T^{6} - 14820 p T^{7} - 1947 p^{2} T^{8} + 979 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 21 T + 180 T^{2} - 1119 T^{3} + 10053 T^{4} - 100416 T^{5} + 788551 T^{6} - 100416 p T^{7} + 10053 p^{2} T^{8} - 1119 p^{3} T^{9} + 180 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 - 6 T + 162 T^{2} - 627 T^{3} + 162 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 31 T + 476 T^{2} - 5741 T^{3} + 62553 T^{4} - 587576 T^{5} + 4781851 T^{6} - 587576 p T^{7} + 62553 p^{2} T^{8} - 5741 p^{3} T^{9} + 476 p^{4} T^{10} - 31 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 6 T + 48 T^{2} + 642 T^{3} + 3018 T^{4} + 35394 T^{5} + 438671 T^{6} + 35394 p T^{7} + 3018 p^{2} T^{8} + 642 p^{3} T^{9} + 48 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 6 T - 150 T^{2} + 506 T^{3} + 17268 T^{4} - 28236 T^{5} - 1220289 T^{6} - 28236 p T^{7} + 17268 p^{2} T^{8} + 506 p^{3} T^{9} - 150 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 17 T + 119 T^{2} + 507 T^{3} + 119 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 + 3 T + 195 T^{2} + 359 T^{3} + 195 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 + 9 T - 114 T^{2} - 351 T^{3} + 13143 T^{4} - 15786 T^{5} - 1414609 T^{6} - 15786 p T^{7} + 13143 p^{2} T^{8} - 351 p^{3} T^{9} - 114 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 20 T + 38 T^{2} - 346 T^{3} + 32058 T^{4} - 183754 T^{5} - 606869 T^{6} - 183754 p T^{7} + 32058 p^{2} T^{8} - 346 p^{3} T^{9} + 38 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
89 \( ( 1 - 12 T + 216 T^{2} - 1425 T^{3} + 216 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 9 T - 66 T^{2} + 2023 T^{3} - 7707 T^{4} - 73950 T^{5} + 1766073 T^{6} - 73950 p T^{7} - 7707 p^{2} T^{8} + 2023 p^{3} T^{9} - 66 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.25086164311522622186102404553, −5.17993430191430297266470912020, −4.81480360222961662271719978010, −4.66893205225040431710220052693, −4.60779012261765606414926685487, −4.42835323597690597869210036442, −4.38947741893270277964301036358, −3.92161640037118546323911048747, −3.79818926427333372453326641029, −3.75576054487435265407405397482, −3.70296531609430440258744601911, −3.50130075459037375592109236441, −3.26166044699315201512058130211, −2.88025037662652357120487895643, −2.81539813650829824982620371699, −2.65689305551791856149339773300, −2.56424208696084448468478371086, −2.45797617606362914769964708672, −1.96787806254207747472817339877, −1.79749541168992469916997131788, −1.62105816068170743284440977895, −1.14243048187873955093498399281, −0.977744371199471365855692125168, −0.823823687012042183320912864673, −0.40774962780391401145471426510, 0.40774962780391401145471426510, 0.823823687012042183320912864673, 0.977744371199471365855692125168, 1.14243048187873955093498399281, 1.62105816068170743284440977895, 1.79749541168992469916997131788, 1.96787806254207747472817339877, 2.45797617606362914769964708672, 2.56424208696084448468478371086, 2.65689305551791856149339773300, 2.81539813650829824982620371699, 2.88025037662652357120487895643, 3.26166044699315201512058130211, 3.50130075459037375592109236441, 3.70296531609430440258744601911, 3.75576054487435265407405397482, 3.79818926427333372453326641029, 3.92161640037118546323911048747, 4.38947741893270277964301036358, 4.42835323597690597869210036442, 4.60779012261765606414926685487, 4.66893205225040431710220052693, 4.81480360222961662271719978010, 5.17993430191430297266470912020, 5.25086164311522622186102404553

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

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