↑

Properties

Label 12-1875e6-1.1-c1e6-0-3
Degree $12$
Conductor $4.345\times 10^{19}$
Sign $1$
Analytic cond. $1.12634\times 10^{7}$
Root an. cond. $3.86936$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s − 4-s − 6·7-s − 8-s + 21·9-s + 3·11-s + 6·12-s − 6·13-s − 13·17-s + 11·19-s + 36·21-s − 13·23-s + 6·24-s − 56·27-s + 6·28-s − 3·29-s − 11·31-s − 32-s − 18·33-s − 21·36-s − 21·37-s + 36·39-s − 41-s − 2·43-s − 3·44-s − 14·47-s − 10·49-s + ⋯
L(s)  = 1  − 3.46·3-s − 1/2·4-s − 2.26·7-s − 0.353·8-s + 7·9-s + 0.904·11-s + 1.73·12-s − 1.66·13-s − 3.15·17-s + 2.52·19-s + 7.85·21-s − 2.71·23-s + 1.22·24-s − 10.7·27-s + 1.13·28-s − 0.557·29-s − 1.97·31-s − 0.176·32-s − 3.13·33-s − 7/2·36-s − 3.45·37-s + 5.76·39-s − 0.156·41-s − 0.304·43-s − 0.452·44-s − 2.04·47-s − 1.42·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 5^{24}\)
Sign: $1$
Analytic conductor: \(1.12634\times 10^{7}\)
Root analytic conductor: \(3.86936\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 3^{6} \cdot 5^{24} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T )^{6} \)
5 \( 1 \)
good2 \( 1 + T^{2} + T^{3} + T^{4} + 3 T^{5} + 11 T^{6} + 3 p T^{7} + p^{2} T^{8} + p^{3} T^{9} + p^{4} T^{10} + p^{6} T^{12} \)
7 \( 1 + 6 T + 46 T^{2} + 185 T^{3} + 822 T^{4} + 2435 T^{5} + 7706 T^{6} + 2435 p T^{7} + 822 p^{2} T^{8} + 185 p^{3} T^{9} + 46 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 3 T + 42 T^{2} - 9 p T^{3} + 870 T^{4} - 1842 T^{5} + 11882 T^{6} - 1842 p T^{7} + 870 p^{2} T^{8} - 9 p^{4} T^{9} + 42 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 6 T + 4 p T^{2} + 217 T^{3} + 1284 T^{4} + 4378 T^{5} + 20303 T^{6} + 4378 p T^{7} + 1284 p^{2} T^{8} + 217 p^{3} T^{9} + 4 p^{5} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 13 T + 117 T^{2} + 40 p T^{3} + 3545 T^{4} + 15273 T^{5} + 67369 T^{6} + 15273 p T^{7} + 3545 p^{2} T^{8} + 40 p^{4} T^{9} + 117 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 11 T + 125 T^{2} - 876 T^{3} + 6052 T^{4} - 30827 T^{5} + 424 p^{2} T^{6} - 30827 p T^{7} + 6052 p^{2} T^{8} - 876 p^{3} T^{9} + 125 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 13 T + 144 T^{2} + 1185 T^{3} + 364 p T^{4} + 49500 T^{5} + 255074 T^{6} + 49500 p T^{7} + 364 p^{3} T^{8} + 1185 p^{3} T^{9} + 144 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 3 T + 133 T^{2} + 262 T^{3} + 8085 T^{4} + 12009 T^{5} + 296107 T^{6} + 12009 p T^{7} + 8085 p^{2} T^{8} + 262 p^{3} T^{9} + 133 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 11 T + 140 T^{2} + 1025 T^{3} + 8226 T^{4} + 46520 T^{5} + 297614 T^{6} + 46520 p T^{7} + 8226 p^{2} T^{8} + 1025 p^{3} T^{9} + 140 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 21 T + 361 T^{2} + 4000 T^{3} + 38967 T^{4} + 7915 p T^{5} + 1990421 T^{6} + 7915 p^{2} T^{7} + 38967 p^{2} T^{8} + 4000 p^{3} T^{9} + 361 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + T + 65 T^{2} + 315 T^{3} + 4526 T^{4} + 16885 T^{5} + 207789 T^{6} + 16885 p T^{7} + 4526 p^{2} T^{8} + 315 p^{3} T^{9} + 65 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 2 T + 167 T^{2} + 256 T^{3} + 13452 T^{4} + 15956 T^{5} + 692036 T^{6} + 15956 p T^{7} + 13452 p^{2} T^{8} + 256 p^{3} T^{9} + 167 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 14 T + 278 T^{2} + 2603 T^{3} + 30072 T^{4} + 211487 T^{5} + 1808494 T^{6} + 211487 p T^{7} + 30072 p^{2} T^{8} + 2603 p^{3} T^{9} + 278 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 23 T + 379 T^{2} + 4310 T^{3} + 40637 T^{4} + 323685 T^{5} + 2441559 T^{6} + 323685 p T^{7} + 40637 p^{2} T^{8} + 4310 p^{3} T^{9} + 379 p^{4} T^{10} + 23 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 9 T + 265 T^{2} - 2264 T^{3} + 32442 T^{4} - 249123 T^{5} + 2397904 T^{6} - 249123 p T^{7} + 32442 p^{2} T^{8} - 2264 p^{3} T^{9} + 265 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 11 T + 227 T^{2} - 1507 T^{3} + 23288 T^{4} - 138147 T^{5} + 1774033 T^{6} - 138147 p T^{7} + 23288 p^{2} T^{8} - 1507 p^{3} T^{9} + 227 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 8 T + 247 T^{2} + 1290 T^{3} + 26550 T^{4} + 92768 T^{5} + 1956344 T^{6} + 92768 p T^{7} + 26550 p^{2} T^{8} + 1290 p^{3} T^{9} + 247 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 8 T + 276 T^{2} + 1095 T^{3} + 27860 T^{4} + 28473 T^{5} + 1889114 T^{6} + 28473 p T^{7} + 27860 p^{2} T^{8} + 1095 p^{3} T^{9} + 276 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 13 T + 364 T^{2} + 3570 T^{3} + 58057 T^{4} + 450270 T^{5} + 5397049 T^{6} + 450270 p T^{7} + 58057 p^{2} T^{8} + 3570 p^{3} T^{9} + 364 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 5 T + 314 T^{2} + 1445 T^{3} + 45010 T^{4} + 190240 T^{5} + 4170310 T^{6} + 190240 p T^{7} + 45010 p^{2} T^{8} + 1445 p^{3} T^{9} + 314 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 20 T + 524 T^{2} - 7048 T^{3} + 106936 T^{4} - 1073796 T^{5} + 11716314 T^{6} - 1073796 p T^{7} + 106936 p^{2} T^{8} - 7048 p^{3} T^{9} + 524 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 4 T + 70 T^{2} + 1649 T^{3} + 9072 T^{4} + 84358 T^{5} + 1538759 T^{6} + 84358 p T^{7} + 9072 p^{2} T^{8} + 1649 p^{3} T^{9} + 70 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 7 T + 367 T^{2} - 1905 T^{3} + 69210 T^{4} - 304027 T^{5} + 8348279 T^{6} - 304027 p T^{7} + 69210 p^{2} T^{8} - 1905 p^{3} T^{9} + 367 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.38430546270427350274679481011, −4.95541045982864586404747110251, −4.92380303850649422651596293986, −4.92097941843277901803372304721, −4.90680700416264119749794950418, −4.74356616171398873793149577855, −4.58986776968159248544434546122, −4.03638071102947295065332022120, −4.02029631003388117466887923934, −3.88639494298388764539754071963, −3.84455481812455450925420404540, −3.77212310755955041120996544645, −3.58551772316829954831444845551, −3.24639760750724379691332985261, −3.10561071954785027872377490694, −2.97530393123205402560298374278, −2.93468732331815436161634602199, −2.29417566796639957595857191229, −2.20779470816680977735835228784, −2.13483728993346478922184714353, −2.03963560125530031525842314784, −1.36883888584412572952703562600, −1.36398071880842826656129497539, −1.35661194388039420385169106014, −1.29757094918371103392098490418, 0, 0, 0, 0, 0, 0, 1.29757094918371103392098490418, 1.35661194388039420385169106014, 1.36398071880842826656129497539, 1.36883888584412572952703562600, 2.03963560125530031525842314784, 2.13483728993346478922184714353, 2.20779470816680977735835228784, 2.29417566796639957595857191229, 2.93468732331815436161634602199, 2.97530393123205402560298374278, 3.10561071954785027872377490694, 3.24639760750724379691332985261, 3.58551772316829954831444845551, 3.77212310755955041120996544645, 3.84455481812455450925420404540, 3.88639494298388764539754071963, 4.02029631003388117466887923934, 4.03638071102947295065332022120, 4.58986776968159248544434546122, 4.74356616171398873793149577855, 4.90680700416264119749794950418, 4.92097941843277901803372304721, 4.92380303850649422651596293986, 4.95541045982864586404747110251, 5.38430546270427350274679481011

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

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