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Properties

Label 12-30e6-1.1-c11e6-0-0
Degree $12$
Conductor $729000000$
Sign $1$
Analytic cond. $1.49988\times 10^{8}$
Root an. cond. $4.80107$
Motivic weight $11$
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  − 3.07e3·4-s − 9.92e3·5-s − 1.77e5·9-s + 1.75e6·11-s + 6.29e6·16-s − 4.06e6·19-s + 3.04e7·20-s + 1.34e8·25-s + 7.17e7·29-s − 1.22e8·31-s + 5.44e8·36-s + 2.00e9·41-s − 5.38e9·44-s + 1.75e9·45-s + 1.13e9·49-s − 1.74e10·55-s − 4.06e9·59-s − 1.23e10·61-s − 1.07e10·64-s + 1.37e10·71-s + 1.25e10·76-s − 1.34e10·79-s − 6.24e10·80-s + 2.09e10·81-s − 1.27e11·89-s + 4.03e10·95-s − 3.10e11·99-s + ⋯
L(s)  = 1  − 3/2·4-s − 1.42·5-s − 9-s + 3.28·11-s + 3/2·16-s − 0.377·19-s + 2.13·20-s + 2.74·25-s + 0.649·29-s − 0.766·31-s + 3/2·36-s + 2.70·41-s − 4.92·44-s + 1.42·45-s + 0.574·49-s − 4.66·55-s − 0.741·59-s − 1.87·61-s − 5/4·64-s + 0.901·71-s + 0.565·76-s − 0.491·79-s − 2.13·80-s + 2/3·81-s − 2.41·89-s + 0.535·95-s − 3.28·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 5^{6}\)
Sign: $1$
Analytic conductor: \(1.49988\times 10^{8}\)
Root analytic conductor: \(4.80107\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 5^{6} ,\ ( \ : [11/2]^{6} ),\ 1 )\)

Particular Values

\(L(6)\) \(\approx\) \(2.418022739\)
\(L(\frac12)\) \(\approx\) \(2.418022739\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + p^{10} T^{2} )^{3} \)
3 \( ( 1 + p^{10} T^{2} )^{3} \)
5 \( 1 + 9926 T - 1429057 p^{2} T^{2} - 10837908 p^{7} T^{3} - 1429057 p^{13} T^{4} + 9926 p^{22} T^{5} + p^{33} T^{6} \)
good7 \( 1 - 1136527506 T^{2} + 6691910885188971759 T^{4} - \)\(63\!\cdots\!04\)\( p^{2} T^{6} + 6691910885188971759 p^{22} T^{8} - 1136527506 p^{44} T^{10} + p^{66} T^{12} \)
11 \( ( 1 - 79700 p T + 624805292933 T^{2} - 280023173614637400 T^{3} + 624805292933 p^{11} T^{4} - 79700 p^{23} T^{5} + p^{33} T^{6} )^{2} \)
13 \( 1 - 2325639202134 T^{2} + \)\(41\!\cdots\!59\)\( T^{4} - \)\(94\!\cdots\!44\)\( T^{6} + \)\(41\!\cdots\!59\)\( p^{22} T^{8} - 2325639202134 p^{44} T^{10} + p^{66} T^{12} \)
17 \( 1 - 152163753945006 T^{2} + \)\(11\!\cdots\!79\)\( T^{4} - \)\(48\!\cdots\!76\)\( T^{6} + \)\(11\!\cdots\!79\)\( p^{22} T^{8} - 152163753945006 p^{44} T^{10} + p^{66} T^{12} \)
19 \( ( 1 + 2034912 T + 277479196963905 T^{2} + \)\(18\!\cdots\!20\)\( T^{3} + 277479196963905 p^{11} T^{4} + 2034912 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
23 \( 1 - 884282120093562 T^{2} + \)\(18\!\cdots\!35\)\( T^{4} - \)\(83\!\cdots\!60\)\( T^{6} + \)\(18\!\cdots\!35\)\( p^{22} T^{8} - 884282120093562 p^{44} T^{10} + p^{66} T^{12} \)
29 \( ( 1 - 35857782 T + 14889529536068895 T^{2} + \)\(35\!\cdots\!60\)\( T^{3} + 14889529536068895 p^{11} T^{4} - 35857782 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
31 \( ( 1 + 61121676 T + 25960618947717885 T^{2} + \)\(57\!\cdots\!00\)\( T^{3} + 25960618947717885 p^{11} T^{4} + 61121676 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
37 \( 1 - 723093277527173190 T^{2} + \)\(26\!\cdots\!07\)\( T^{4} - \)\(57\!\cdots\!20\)\( T^{6} + \)\(26\!\cdots\!07\)\( p^{22} T^{8} - 723093277527173190 p^{44} T^{10} + p^{66} T^{12} \)
41 \( ( 1 - 1003384282 T + 880908677252228231 T^{2} - \)\(97\!\cdots\!08\)\( T^{3} + 880908677252228231 p^{11} T^{4} - 1003384282 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
43 \( 1 - 3918537335019954690 T^{2} + \)\(73\!\cdots\!47\)\( T^{4} - \)\(85\!\cdots\!20\)\( T^{6} + \)\(73\!\cdots\!47\)\( p^{22} T^{8} - 3918537335019954690 p^{44} T^{10} + p^{66} T^{12} \)
47 \( 1 - 1932557424779667018 T^{2} + \)\(11\!\cdots\!35\)\( T^{4} - \)\(12\!\cdots\!40\)\( T^{6} + \)\(11\!\cdots\!35\)\( p^{22} T^{8} - 1932557424779667018 p^{44} T^{10} + p^{66} T^{12} \)
53 \( 1 - 30517521498331931190 T^{2} + \)\(41\!\cdots\!27\)\( T^{4} - \)\(40\!\cdots\!20\)\( T^{6} + \)\(41\!\cdots\!27\)\( p^{22} T^{8} - 30517521498331931190 p^{44} T^{10} + p^{66} T^{12} \)
59 \( ( 1 + 2034730100 T + 66824599482384501077 T^{2} + \)\(64\!\cdots\!00\)\( T^{3} + 66824599482384501077 p^{11} T^{4} + 2034730100 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
61 \( ( 1 + 6193233330 T + 74414539314738778683 T^{2} + \)\(19\!\cdots\!60\)\( T^{3} + 74414539314738778683 p^{11} T^{4} + 6193233330 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
67 \( 1 - \)\(62\!\cdots\!46\)\( T^{2} + \)\(17\!\cdots\!39\)\( T^{4} - \)\(27\!\cdots\!56\)\( T^{6} + \)\(17\!\cdots\!39\)\( p^{22} T^{8} - \)\(62\!\cdots\!46\)\( p^{44} T^{10} + p^{66} T^{12} \)
71 \( ( 1 - 6851752800 T + \)\(58\!\cdots\!13\)\( T^{2} - \)\(23\!\cdots\!00\)\( T^{3} + \)\(58\!\cdots\!13\)\( p^{11} T^{4} - 6851752800 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
73 \( 1 - \)\(87\!\cdots\!74\)\( T^{2} + \)\(54\!\cdots\!79\)\( T^{4} - \)\(19\!\cdots\!04\)\( T^{6} + \)\(54\!\cdots\!79\)\( p^{22} T^{8} - \)\(87\!\cdots\!74\)\( p^{44} T^{10} + p^{66} T^{12} \)
79 \( ( 1 + 6719163876 T + \)\(57\!\cdots\!29\)\( T^{2} - \)\(19\!\cdots\!04\)\( T^{3} + \)\(57\!\cdots\!29\)\( p^{11} T^{4} + 6719163876 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
83 \( 1 + \)\(42\!\cdots\!66\)\( T^{2} + \)\(10\!\cdots\!19\)\( T^{4} + \)\(15\!\cdots\!96\)\( T^{6} + \)\(10\!\cdots\!19\)\( p^{22} T^{8} + \)\(42\!\cdots\!66\)\( p^{44} T^{10} + p^{66} T^{12} \)
89 \( ( 1 + 63718294546 T + \)\(92\!\cdots\!39\)\( T^{2} + \)\(34\!\cdots\!56\)\( T^{3} + \)\(92\!\cdots\!39\)\( p^{11} T^{4} + 63718294546 p^{22} T^{5} + p^{33} T^{6} )^{2} \)
97 \( 1 - \)\(27\!\cdots\!30\)\( T^{2} + \)\(29\!\cdots\!27\)\( T^{4} + \)\(11\!\cdots\!60\)\( T^{6} + \)\(29\!\cdots\!27\)\( p^{22} T^{8} - \)\(27\!\cdots\!30\)\( p^{44} T^{10} + p^{66} T^{12} \)
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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

−7.15027224029294388115762775604, −6.93222580675907995527927061340, −6.68432554619473876588523138812, −6.63866991311177738180284371027, −6.21851511006189160768443984141, −5.76790943958575966796842659975, −5.70455349463337020833320844485, −5.55654640413340140059846918037, −4.91196069549311910721823771022, −4.57492410224302979885262832582, −4.45834585138622326845891430848, −4.36026020743116458999998859313, −4.04740407205722960395636432926, −3.65826123777186218779754850024, −3.62032084465942930265623517185, −3.11070899272506628444719819268, −2.95858675657909595509156928211, −2.69245310748741044106504448458, −1.92474978127773806470020071102, −1.80670303825935939790317955377, −1.23802608673735608325024454052, −1.04982967352556303803832571432, −0.71458673224099920920202598416, −0.59373097540416759429421709920, −0.22460499025096899667949478958, 0.22460499025096899667949478958, 0.59373097540416759429421709920, 0.71458673224099920920202598416, 1.04982967352556303803832571432, 1.23802608673735608325024454052, 1.80670303825935939790317955377, 1.92474978127773806470020071102, 2.69245310748741044106504448458, 2.95858675657909595509156928211, 3.11070899272506628444719819268, 3.62032084465942930265623517185, 3.65826123777186218779754850024, 4.04740407205722960395636432926, 4.36026020743116458999998859313, 4.45834585138622326845891430848, 4.57492410224302979885262832582, 4.91196069549311910721823771022, 5.55654640413340140059846918037, 5.70455349463337020833320844485, 5.76790943958575966796842659975, 6.21851511006189160768443984141, 6.63866991311177738180284371027, 6.68432554619473876588523138812, 6.93222580675907995527927061340, 7.15027224029294388115762775604

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

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