Properties

Label 12-21e12-1.1-c5e6-0-0
Degree $12$
Conductor $7.356\times 10^{15}$
Sign $1$
Analytic cond. $1.25197\times 10^{11}$
Root an. cond. $8.41006$
Motivic weight $5$
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  − 5·4-s − 154·13-s − 969·16-s + 9.42e3·19-s − 5.61e3·25-s + 2.34e4·31-s + 1.81e4·37-s − 4.36e4·43-s + 770·52-s − 1.61e4·61-s + 2.64e4·64-s − 1.44e5·67-s − 1.00e5·73-s − 4.71e4·76-s − 1.01e5·79-s + 4.33e5·97-s + 2.80e4·100-s + 1.45e5·103-s − 2.43e5·109-s − 6.37e4·121-s − 1.17e5·124-s + 127-s + 131-s + 137-s + 139-s − 9.09e4·148-s + 149-s + ⋯
L(s)  = 1  − 0.156·4-s − 0.252·13-s − 0.946·16-s + 5.98·19-s − 1.79·25-s + 4.37·31-s + 2.18·37-s − 3.60·43-s + 0.0394·52-s − 0.555·61-s + 0.808·64-s − 3.93·67-s − 2.19·73-s − 0.935·76-s − 1.83·79-s + 4.67·97-s + 0.280·100-s + 1.34·103-s − 1.96·109-s − 0.395·121-s − 0.683·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s − 0.341·148-s + 3.69e−6·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(1.25197\times 10^{11}\)
Root analytic conductor: \(8.41006\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 7^{12} ,\ ( \ : [5/2]^{6} ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(2.200113904\)
\(L(\frac12)\) \(\approx\) \(2.200113904\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + 5 T^{2} + 497 p T^{4} - 2083 p^{3} T^{6} + 497 p^{11} T^{8} + 5 p^{20} T^{10} + p^{30} T^{12} \)
5 \( 1 + 5612 T^{2} + 33696448 T^{4} + 105800333314 T^{6} + 33696448 p^{10} T^{8} + 5612 p^{20} T^{10} + p^{30} T^{12} \)
11 \( 1 + 63728 T^{2} + 42247579024 T^{4} + 1529842075323718 T^{6} + 42247579024 p^{10} T^{8} + 63728 p^{20} T^{10} + p^{30} T^{12} \)
13 \( ( 1 + 77 T + 329903 T^{2} - 2901514 T^{3} + 329903 p^{5} T^{4} + 77 p^{10} T^{5} + p^{15} T^{6} )^{2} \)
17 \( 1 + 366750 p T^{2} + 18438724931679 T^{4} + 32803889999214179716 T^{6} + 18438724931679 p^{10} T^{8} + 366750 p^{21} T^{10} + p^{30} T^{12} \)
19 \( ( 1 - 4711 T + 14724701 T^{2} - 1423351594 p T^{3} + 14724701 p^{5} T^{4} - 4711 p^{10} T^{5} + p^{15} T^{6} )^{2} \)
23 \( 1 + 28426626 T^{2} + 379254398764527 T^{4} + \)\(30\!\cdots\!04\)\( T^{6} + 379254398764527 p^{10} T^{8} + 28426626 p^{20} T^{10} + p^{30} T^{12} \)
29 \( 1 + 56877260 T^{2} + 2153523833778304 T^{4} + \)\(50\!\cdots\!14\)\( T^{6} + 2153523833778304 p^{10} T^{8} + 56877260 p^{20} T^{10} + p^{30} T^{12} \)
31 \( ( 1 - 11711 T + 91157164 T^{2} - 584745089971 T^{3} + 91157164 p^{5} T^{4} - 11711 p^{10} T^{5} + p^{15} T^{6} )^{2} \)
37 \( ( 1 - 9091 T + 131490511 T^{2} - 1228462202570 T^{3} + 131490511 p^{5} T^{4} - 9091 p^{10} T^{5} + p^{15} T^{6} )^{2} \)
41 \( 1 - 32763274 T^{2} + 36122988699522367 T^{4} - \)\(94\!\cdots\!80\)\( T^{6} + 36122988699522367 p^{10} T^{8} - 32763274 p^{20} T^{10} + p^{30} T^{12} \)
43 \( ( 1 + 21843 T + 434197953 T^{2} + 4778588910994 T^{3} + 434197953 p^{5} T^{4} + 21843 p^{10} T^{5} + p^{15} T^{6} )^{2} \)
47 \( 1 + 762615162 T^{2} + 305120186364825711 T^{4} + \)\(81\!\cdots\!00\)\( T^{6} + 305120186364825711 p^{10} T^{8} + 762615162 p^{20} T^{10} + p^{30} T^{12} \)
53 \( 1 + 1842711660 T^{2} + 1522579120817053824 T^{4} + \)\(77\!\cdots\!34\)\( T^{6} + 1522579120817053824 p^{10} T^{8} + 1842711660 p^{20} T^{10} + p^{30} T^{12} \)
59 \( 1 - 405776592 T^{2} + 1198265548187923920 T^{4} - \)\(27\!\cdots\!62\)\( T^{6} + 1198265548187923920 p^{10} T^{8} - 405776592 p^{20} T^{10} + p^{30} T^{12} \)
61 \( ( 1 + 8078 T + 2037780407 T^{2} + 8299164526604 T^{3} + 2037780407 p^{5} T^{4} + 8078 p^{10} T^{5} + p^{15} T^{6} )^{2} \)
67 \( ( 1 + 72325 T + 5554510253 T^{2} + 202408145385274 T^{3} + 5554510253 p^{5} T^{4} + 72325 p^{10} T^{5} + p^{15} T^{6} )^{2} \)
71 \( 1 + 7343984298 T^{2} + 25986157270151134623 T^{4} + \)\(57\!\cdots\!08\)\( T^{6} + 25986157270151134623 p^{10} T^{8} + 7343984298 p^{20} T^{10} + p^{30} T^{12} \)
73 \( ( 1 + 50029 T + 6702442195 T^{2} + 208365222113006 T^{3} + 6702442195 p^{5} T^{4} + 50029 p^{10} T^{5} + p^{15} T^{6} )^{2} \)
79 \( ( 1 + 50997 T + 5666795124 T^{2} + 202281745014889 T^{3} + 5666795124 p^{5} T^{4} + 50997 p^{10} T^{5} + p^{15} T^{6} )^{2} \)
83 \( 1 + 15907224432 T^{2} + \)\(11\!\cdots\!44\)\( T^{4} + \)\(52\!\cdots\!98\)\( T^{6} + \)\(11\!\cdots\!44\)\( p^{10} T^{8} + 15907224432 p^{20} T^{10} + p^{30} T^{12} \)
89 \( 1 + 30683976398 T^{2} + \)\(40\!\cdots\!03\)\( T^{4} + \)\(29\!\cdots\!00\)\( T^{6} + \)\(40\!\cdots\!03\)\( p^{10} T^{8} + 30683976398 p^{20} T^{10} + p^{30} T^{12} \)
97 \( ( 1 - 216524 T + 31765510816 T^{2} - 3171927842758234 T^{3} + 31765510816 p^{5} T^{4} - 216524 p^{10} T^{5} + p^{15} T^{6} )^{2} \)
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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

−5.29943269706303214340559992123, −4.89478014242335983722153520409, −4.68607916599759017913458556336, −4.61111386690822090204745721620, −4.53247075827934243432998131661, −4.26999520672535621404230216818, −4.23867885000401851003738294648, −3.74687998727126335463621090357, −3.56566345943164200006791294229, −3.27093907264410985878642140674, −3.25990725986850750993033739667, −3.02711212529684676516252857779, −3.01081359549177594121496351698, −2.66342565217714952171284656452, −2.61351583089249290058416606194, −2.32759299888981152236185904136, −1.91431612029192322944467487077, −1.58684139249484044742526990129, −1.49655450083800899060553107835, −1.40400911544250674092077364003, −1.04029428528378407579072550014, −0.871785836773034726775839000113, −0.64463934952811272271827424688, −0.51288441384034722742801495460, −0.094662332482458449387151588158, 0.094662332482458449387151588158, 0.51288441384034722742801495460, 0.64463934952811272271827424688, 0.871785836773034726775839000113, 1.04029428528378407579072550014, 1.40400911544250674092077364003, 1.49655450083800899060553107835, 1.58684139249484044742526990129, 1.91431612029192322944467487077, 2.32759299888981152236185904136, 2.61351583089249290058416606194, 2.66342565217714952171284656452, 3.01081359549177594121496351698, 3.02711212529684676516252857779, 3.25990725986850750993033739667, 3.27093907264410985878642140674, 3.56566345943164200006791294229, 3.74687998727126335463621090357, 4.23867885000401851003738294648, 4.26999520672535621404230216818, 4.53247075827934243432998131661, 4.61111386690822090204745721620, 4.68607916599759017913458556336, 4.89478014242335983722153520409, 5.29943269706303214340559992123

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

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