Properties

Label 12-1280e6-1.1-c2e6-0-0
Degree $12$
Conductor $4.398\times 10^{18}$
Sign $1$
Analytic cond. $1.79999\times 10^{9}$
Root an. cond. $5.90571$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·5-s − 12·7-s + 18·9-s − 8·11-s + 24·19-s + 68·23-s + 37·25-s + 96·35-s + 208·37-s + 68·41-s − 144·45-s + 268·47-s − 106·49-s − 64·53-s + 64·55-s − 360·59-s − 216·63-s + 96·77-s + 181·81-s − 76·89-s − 192·95-s − 144·99-s − 124·103-s − 544·115-s − 134·121-s − 160·125-s + 127-s + ⋯
L(s)  = 1  − 8/5·5-s − 1.71·7-s + 2·9-s − 0.727·11-s + 1.26·19-s + 2.95·23-s + 1.47·25-s + 2.74·35-s + 5.62·37-s + 1.65·41-s − 3.19·45-s + 5.70·47-s − 2.16·49-s − 1.20·53-s + 1.16·55-s − 6.10·59-s − 3.42·63-s + 1.24·77-s + 2.23·81-s − 0.853·89-s − 2.02·95-s − 1.45·99-s − 1.20·103-s − 4.73·115-s − 1.10·121-s − 1.27·125-s + 0.00787·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{48} \cdot 5^{6}\)
Sign: $1$
Analytic conductor: \(1.79999\times 10^{9}\)
Root analytic conductor: \(5.90571\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1280} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{48} \cdot 5^{6} ,\ ( \ : [1]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.09450023079\)
\(L(\frac12)\) \(\approx\) \(0.09450023079\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + 8 T + 27 T^{2} + 16 p T^{3} + 27 p^{2} T^{4} + 8 p^{4} T^{5} + p^{6} T^{6} \)
good3 \( 1 - 2 p^{2} T^{2} + 143 T^{4} - 1052 T^{6} + 143 p^{4} T^{8} - 2 p^{10} T^{10} + p^{12} T^{12} \)
7 \( ( 1 + 6 T + 107 T^{2} + 596 T^{3} + 107 p^{2} T^{4} + 6 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
11 \( ( 1 + 4 T + 91 T^{2} - 632 T^{3} + 91 p^{2} T^{4} + 4 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
13 \( ( 1 + 23 p T^{2} - 64 p T^{3} + 23 p^{3} T^{4} + p^{6} T^{6} )^{2} \)
17 \( 1 - 38 p T^{2} + 265423 T^{4} - 87226900 T^{6} + 265423 p^{4} T^{8} - 38 p^{9} T^{10} + p^{12} T^{12} \)
19 \( ( 1 - 12 T + 299 T^{2} - 12056 T^{3} + 299 p^{2} T^{4} - 12 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
23 \( ( 1 - 34 T + 1435 T^{2} - 26812 T^{3} + 1435 p^{2} T^{4} - 34 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
29 \( 1 - 2666 T^{2} + 3408319 T^{4} - 3146722508 T^{6} + 3408319 p^{4} T^{8} - 2666 p^{8} T^{10} + p^{12} T^{12} \)
31 \( 1 - 3206 T^{2} + 5912719 T^{4} - 6798206228 T^{6} + 5912719 p^{4} T^{8} - 3206 p^{8} T^{10} + p^{12} T^{12} \)
37 \( ( 1 - 104 T + 6811 T^{2} - 305552 T^{3} + 6811 p^{2} T^{4} - 104 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
41 \( ( 1 - 34 T + 4943 T^{2} - 109308 T^{3} + 4943 p^{2} T^{4} - 34 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
43 \( 1 - 6834 T^{2} + 22234799 T^{4} - 48099402332 T^{6} + 22234799 p^{4} T^{8} - 6834 p^{8} T^{10} + p^{12} T^{12} \)
47 \( ( 1 - 134 T + 11451 T^{2} - 614228 T^{3} + 11451 p^{2} T^{4} - 134 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
53 \( ( 1 + 32 T + 2459 T^{2} - 44544 T^{3} + 2459 p^{2} T^{4} + 32 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
59 \( ( 1 + 180 T + 20411 T^{2} + 1412584 T^{3} + 20411 p^{2} T^{4} + 180 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
61 \( 1 - 16362 T^{2} + 130073663 T^{4} - 611308223948 T^{6} + 130073663 p^{4} T^{8} - 16362 p^{8} T^{10} + p^{12} T^{12} \)
67 \( 1 - 12754 T^{2} + 78010639 T^{4} - 366668429212 T^{6} + 78010639 p^{4} T^{8} - 12754 p^{8} T^{10} + p^{12} T^{12} \)
71 \( 1 - 21862 T^{2} + 231858863 T^{4} - 1468469748948 T^{6} + 231858863 p^{4} T^{8} - 21862 p^{8} T^{10} + p^{12} T^{12} \)
73 \( 1 - 27558 T^{2} + 338289263 T^{4} - 2339843433812 T^{6} + 338289263 p^{4} T^{8} - 27558 p^{8} T^{10} + p^{12} T^{12} \)
79 \( 1 - 11590 T^{2} + 98296655 T^{4} - 773190011028 T^{6} + 98296655 p^{4} T^{8} - 11590 p^{8} T^{10} + p^{12} T^{12} \)
83 \( 1 - 17170 T^{2} + 191959631 T^{4} - 1435306239516 T^{6} + 191959631 p^{4} T^{8} - 17170 p^{8} T^{10} + p^{12} T^{12} \)
89 \( ( 1 + 38 T + 9823 T^{2} + 446996 T^{3} + 9823 p^{2} T^{4} + 38 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
97 \( 1 - 48966 T^{2} + 1047505295 T^{4} - 12708004801940 T^{6} + 1047505295 p^{4} T^{8} - 48966 p^{8} T^{10} + p^{12} T^{12} \)
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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

−4.88522905638952787187368864933, −4.57529377480404553460913302650, −4.49847821680769772523495663655, −4.34691126194371572584429275957, −4.22365439671619082628461564887, −4.14121809926537490557902071867, −4.11497535415414476400612336919, −3.64972700431387859873247793502, −3.61763373648666297354780994524, −3.37422146522448551971967246030, −3.13530260843972310627578719396, −2.94393143741009669378018399596, −2.91687088358048661067811036367, −2.87271072677316796165710519301, −2.59731229183975550530958239957, −2.41945800842489072084990860868, −2.30376524404685983250621265921, −1.77746726963888051164089044163, −1.43300233914330201948610735681, −1.32058782344197500614005172044, −1.19771029690457182847580116795, −0.828865936131464766320126838151, −0.790508063346870422406089959621, −0.55544139502132270568153241795, −0.03014689718918358038164487782, 0.03014689718918358038164487782, 0.55544139502132270568153241795, 0.790508063346870422406089959621, 0.828865936131464766320126838151, 1.19771029690457182847580116795, 1.32058782344197500614005172044, 1.43300233914330201948610735681, 1.77746726963888051164089044163, 2.30376524404685983250621265921, 2.41945800842489072084990860868, 2.59731229183975550530958239957, 2.87271072677316796165710519301, 2.91687088358048661067811036367, 2.94393143741009669378018399596, 3.13530260843972310627578719396, 3.37422146522448551971967246030, 3.61763373648666297354780994524, 3.64972700431387859873247793502, 4.11497535415414476400612336919, 4.14121809926537490557902071867, 4.22365439671619082628461564887, 4.34691126194371572584429275957, 4.49847821680769772523495663655, 4.57529377480404553460913302650, 4.88522905638952787187368864933

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

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