Properties

Label 12-1280e6-1.1-c2e6-0-1
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 + 12/7·7-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 + 16/5·45-s − 5.70·47-s − 2.16·49-s + 1.20·53-s − 1.16·55-s − 6.10·59-s + 24/7·63-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{48} \cdot 5^{6} ,\ ( \ : [1]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.02868727057\)
\(L(\frac12)\) \(\approx\) \(0.02868727057\)
\(L(2)\) 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

−4.99237264437129961382340521867, −4.98779326772182622027868311130, −4.83303801794442923719672119620, −4.32694565682667527959953699755, −4.30288920366147991008840810365, −4.17652895461342884484464628837, −3.95239571696187477377044250560, −3.65212112923056039087956924470, −3.53601527734132041251275146113, −3.45797908504736640814738619136, −3.14414798973591705847732279236, −3.12721208664140285417557556877, −2.94921957299259647207245325724, −2.61194816498120914167932719397, −2.31812204827666937144677414806, −2.11215979387564200042005574392, −2.00235040637490813151509065082, −1.64772870790188416352749620370, −1.59095187298666488401002657101, −1.54529940626650732875892581972, −1.51076435273980806015930126011, −1.31189940609389035699438412269, −0.941084134509783320182373428544, −0.11203873701926383876053261292, −0.04093904014922065995778438076, 0.04093904014922065995778438076, 0.11203873701926383876053261292, 0.941084134509783320182373428544, 1.31189940609389035699438412269, 1.51076435273980806015930126011, 1.54529940626650732875892581972, 1.59095187298666488401002657101, 1.64772870790188416352749620370, 2.00235040637490813151509065082, 2.11215979387564200042005574392, 2.31812204827666937144677414806, 2.61194816498120914167932719397, 2.94921957299259647207245325724, 3.12721208664140285417557556877, 3.14414798973591705847732279236, 3.45797908504736640814738619136, 3.53601527734132041251275146113, 3.65212112923056039087956924470, 3.95239571696187477377044250560, 4.17652895461342884484464628837, 4.30288920366147991008840810365, 4.32694565682667527959953699755, 4.83303801794442923719672119620, 4.98779326772182622027868311130, 4.99237264437129961382340521867

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

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