Properties

Label 12-1520e6-1.1-c1e6-0-6
Degree $12$
Conductor $1.233\times 10^{19}$
Sign $1$
Analytic cond. $3.19686\times 10^{6}$
Root an. cond. $3.48385$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3·5-s − 4·7-s + 3·9-s + 10·11-s + 15·13-s + 3·15-s − 17-s − 4·21-s + 4·23-s + 3·25-s + 10·27-s + 2·29-s + 2·31-s + 10·33-s − 12·35-s − 4·37-s + 15·39-s + 2·41-s − 43-s + 9·45-s + 6·47-s − 20·49-s − 51-s − 11·53-s + 30·55-s + 6·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.34·5-s − 1.51·7-s + 9-s + 3.01·11-s + 4.16·13-s + 0.774·15-s − 0.242·17-s − 0.872·21-s + 0.834·23-s + 3/5·25-s + 1.92·27-s + 0.371·29-s + 0.359·31-s + 1.74·33-s − 2.02·35-s − 0.657·37-s + 2.40·39-s + 0.312·41-s − 0.152·43-s + 1.34·45-s + 0.875·47-s − 2.85·49-s − 0.140·51-s − 1.51·53-s + 4.04·55-s + 0.781·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 5^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(3.19686\times 10^{6}\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 5^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(22.38257956\)
\(L(\frac12)\) \(\approx\) \(22.38257956\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( ( 1 - T + T^{2} )^{3} \)
19 \( 1 - 7 p T^{3} + p^{3} T^{6} \)
good3 \( 1 - T - 2 T^{2} - 5 T^{3} + T^{4} + 4 p T^{5} + 19 T^{6} + 4 p^{2} T^{7} + p^{2} T^{8} - 5 p^{3} T^{9} - 2 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
7 \( ( 1 + 2 T + 16 T^{2} + 29 T^{3} + 16 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 - 5 T + 35 T^{2} - 109 T^{3} + 35 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( ( 1 - 7 T + p T^{2} )^{3}( 1 + 2 T + p T^{2} )^{3} \)
17 \( 1 + T - 6 T^{2} - 47 T^{3} - 97 T^{4} + 240 T^{5} + 9433 T^{6} + 240 p T^{7} - 97 p^{2} T^{8} - 47 p^{3} T^{9} - 6 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 4 T - 14 T^{2} + 150 T^{3} - 330 T^{4} - 646 T^{5} + 13395 T^{6} - 646 p T^{7} - 330 p^{2} T^{8} + 150 p^{3} T^{9} - 14 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 2 T - 78 T^{2} + 70 T^{3} + 4112 T^{4} - 1764 T^{5} - 135893 T^{6} - 1764 p T^{7} + 4112 p^{2} T^{8} + 70 p^{3} T^{9} - 78 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 - T + 87 T^{2} - 55 T^{3} + 87 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( ( 1 + 2 T - 8 T^{2} - 79 T^{3} - 8 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 - 2 T - 76 T^{2} + 94 T^{3} + 2866 T^{4} - 402 T^{5} - 115153 T^{6} - 402 p T^{7} + 2866 p^{2} T^{8} + 94 p^{3} T^{9} - 76 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + T - 84 T^{2} + 155 T^{3} + 3605 T^{4} - 8436 T^{5} - 152285 T^{6} - 8436 p T^{7} + 3605 p^{2} T^{8} + 155 p^{3} T^{9} - 84 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 6 T - 98 T^{2} + 226 T^{3} + 8568 T^{4} - 6688 T^{5} - 450585 T^{6} - 6688 p T^{7} + 8568 p^{2} T^{8} + 226 p^{3} T^{9} - 98 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 11 T + 4 T^{2} - 423 T^{3} - 1917 T^{4} - 5488 T^{5} - 36627 T^{6} - 5488 p T^{7} - 1917 p^{2} T^{8} - 423 p^{3} T^{9} + 4 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 6 T - 134 T^{2} + 298 T^{3} + 14916 T^{4} - 12556 T^{5} - 985377 T^{6} - 12556 p T^{7} + 14916 p^{2} T^{8} + 298 p^{3} T^{9} - 134 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 9 T - 53 T^{2} + 892 T^{3} + 341 T^{4} - 26923 T^{5} + 157646 T^{6} - 26923 p T^{7} + 341 p^{2} T^{8} + 892 p^{3} T^{9} - 53 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 20 T + 91 T^{2} + 644 T^{3} + 19418 T^{4} + 116972 T^{5} + 70523 T^{6} + 116972 p T^{7} + 19418 p^{2} T^{8} + 644 p^{3} T^{9} + 91 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 29 T + 392 T^{2} + 3851 T^{3} + 36757 T^{4} + 355872 T^{5} + 3184895 T^{6} + 355872 p T^{7} + 36757 p^{2} T^{8} + 3851 p^{3} T^{9} + 392 p^{4} T^{10} + 29 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 22 T + 148 T^{2} - 814 T^{3} + 16010 T^{4} - 143110 T^{5} + 774911 T^{6} - 143110 p T^{7} + 16010 p^{2} T^{8} - 814 p^{3} T^{9} + 148 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 24 T + 223 T^{2} + 1384 T^{3} + 11666 T^{4} + 62240 T^{5} + 107087 T^{6} + 62240 p T^{7} + 11666 p^{2} T^{8} + 1384 p^{3} T^{9} + 223 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 - 3 T + 195 T^{2} - 575 T^{3} + 195 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 14 T - 35 T^{2} + 1638 T^{3} - 1302 T^{4} - 97958 T^{5} + 1042389 T^{6} - 97958 p T^{7} - 1302 p^{2} T^{8} + 1638 p^{3} T^{9} - 35 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 7 T - 176 T^{2} - 899 T^{3} + 20141 T^{4} + 38638 T^{5} - 2016151 T^{6} + 38638 p T^{7} + 20141 p^{2} T^{8} - 899 p^{3} T^{9} - 176 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.02946612469619260493867962738, −4.70997868820301282381402693261, −4.42340920034426846235560294137, −4.40454429756971503093498159376, −4.35521836110666599235972381823, −4.22879325221765461172843269847, −3.92378757235848158182085326267, −3.77475849069243258570198178211, −3.68091204074545191807288629790, −3.61282171090185466277545433525, −3.23177797785476918331971850757, −3.22361899889921561813444484197, −3.01256951842232508180883009105, −2.95617078311195256252836179522, −2.79829826178496471825122020489, −2.60937356737343603659511023245, −1.99167756695636966760785121610, −1.96800834543383774011744135318, −1.57928678146129614105410370352, −1.56318954950286454183937721118, −1.54883857646813537268135246014, −1.34408463933441205796984292032, −0.944318733098938871439714890422, −0.803641513795898249676624971418, −0.47292153138323557865301189351, 0.47292153138323557865301189351, 0.803641513795898249676624971418, 0.944318733098938871439714890422, 1.34408463933441205796984292032, 1.54883857646813537268135246014, 1.56318954950286454183937721118, 1.57928678146129614105410370352, 1.96800834543383774011744135318, 1.99167756695636966760785121610, 2.60937356737343603659511023245, 2.79829826178496471825122020489, 2.95617078311195256252836179522, 3.01256951842232508180883009105, 3.22361899889921561813444484197, 3.23177797785476918331971850757, 3.61282171090185466277545433525, 3.68091204074545191807288629790, 3.77475849069243258570198178211, 3.92378757235848158182085326267, 4.22879325221765461172843269847, 4.35521836110666599235972381823, 4.40454429756971503093498159376, 4.42340920034426846235560294137, 4.70997868820301282381402693261, 5.02946612469619260493867962738

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

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