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Properties

Label 12-592e6-1.1-c1e6-0-3
Degree $12$
Conductor $4.305\times 10^{16}$
Sign $1$
Analytic cond. $11158.1$
Root an. cond. $2.17419$
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  + 3·3-s + 3·5-s − 6·7-s + 3·9-s + 9·11-s + 9·15-s − 9·19-s − 18·21-s + 15·23-s + 15·25-s − 27-s + 18·31-s + 27·33-s − 18·35-s + 9·37-s + 6·41-s − 12·43-s + 9·45-s + 3·47-s + 18·49-s − 18·53-s + 27·55-s − 27·57-s + 6·59-s − 12·61-s − 18·63-s + 3·67-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.34·5-s − 2.26·7-s + 9-s + 2.71·11-s + 2.32·15-s − 2.06·19-s − 3.92·21-s + 3.12·23-s + 3·25-s − 0.192·27-s + 3.23·31-s + 4.70·33-s − 3.04·35-s + 1.47·37-s + 0.937·41-s − 1.82·43-s + 1.34·45-s + 0.437·47-s + 18/7·49-s − 2.47·53-s + 3.64·55-s − 3.57·57-s + 0.781·59-s − 1.53·61-s − 2.26·63-s + 0.366·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 37^{6}\)
Sign: $1$
Analytic conductor: \(11158.1\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 37^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(11.08112110\)
\(L(\frac12)\) \(\approx\) \(11.08112110\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 - 9 T + 54 T^{2} - 305 T^{3} + 54 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
good3 \( 1 - p T + 2 p T^{2} - 8 T^{3} + p T^{4} + 7 p T^{5} - 53 T^{6} + 7 p^{2} T^{7} + p^{3} T^{8} - 8 p^{3} T^{9} + 2 p^{5} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
5 \( 1 - 3 T - 6 T^{2} + 38 T^{3} - 51 T^{4} - 117 T^{5} + 581 T^{6} - 117 p T^{7} - 51 p^{2} T^{8} + 38 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 6 T + 18 T^{2} + 51 T^{3} + 99 T^{4} + 69 T^{5} - 19 T^{6} + 69 p T^{7} + 99 p^{2} T^{8} + 51 p^{3} T^{9} + 18 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 9 T + 24 T^{2} - 83 T^{3} + 687 T^{4} - 2058 T^{5} + 3347 T^{6} - 2058 p T^{7} + 687 p^{2} T^{8} - 83 p^{3} T^{9} + 24 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 36 T^{2} + 27 T^{3} + 63 p T^{4} + 549 T^{5} + 12977 T^{6} + 549 p T^{7} + 63 p^{3} T^{8} + 27 p^{3} T^{9} + 36 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 + 126 T^{3} + 10963 T^{6} + 126 p^{3} T^{9} + p^{6} T^{12} \)
19 \( 1 + 9 T + 63 T^{2} + 295 T^{3} + 1188 T^{4} + 3510 T^{5} + 11397 T^{6} + 3510 p T^{7} + 1188 p^{2} T^{8} + 295 p^{3} T^{9} + 63 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 15 T + 102 T^{2} - 459 T^{3} + 1905 T^{4} - 8304 T^{5} + 37591 T^{6} - 8304 p T^{7} + 1905 p^{2} T^{8} - 459 p^{3} T^{9} + 102 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 24 T^{2} - 342 T^{3} - 120 T^{4} + 4104 T^{5} + 61315 T^{6} + 4104 p T^{7} - 120 p^{2} T^{8} - 342 p^{3} T^{9} - 24 p^{4} T^{10} + p^{6} T^{12} \)
31 \( ( 1 - 9 T + 117 T^{2} - 575 T^{3} + 117 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 - 6 T + 36 T^{2} + 54 T^{3} - 288 T^{4} + 5232 T^{5} + 44551 T^{6} + 5232 p T^{7} - 288 p^{2} T^{8} + 54 p^{3} T^{9} + 36 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( ( 1 + 6 T + 48 T^{2} + 49 T^{3} + 48 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 3 T - 87 T^{2} + 310 T^{3} + 81 p T^{4} - 8259 T^{5} - 155986 T^{6} - 8259 p T^{7} + 81 p^{3} T^{8} + 310 p^{3} T^{9} - 87 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 18 T + 144 T^{2} + 549 T^{3} - 4113 T^{4} - 79569 T^{5} - 665387 T^{6} - 79569 p T^{7} - 4113 p^{2} T^{8} + 549 p^{3} T^{9} + 144 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 6 T + 120 T^{2} - 83 T^{3} + 3477 T^{4} + 46503 T^{5} + 35645 T^{6} + 46503 p T^{7} + 3477 p^{2} T^{8} - 83 p^{3} T^{9} + 120 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 12 T + 48 T^{2} + 220 T^{3} - 4320 T^{4} - 57240 T^{5} - 269733 T^{6} - 57240 p T^{7} - 4320 p^{2} T^{8} + 220 p^{3} T^{9} + 48 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 3 T - 36 T^{2} + 1140 T^{3} - 4194 T^{4} - 29397 T^{5} + 912401 T^{6} - 29397 p T^{7} - 4194 p^{2} T^{8} + 1140 p^{3} T^{9} - 36 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 6 T - 144 T^{2} + 1080 T^{3} + 2700 T^{4} - 46464 T^{5} + 353557 T^{6} - 46464 p T^{7} + 2700 p^{2} T^{8} + 1080 p^{3} T^{9} - 144 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
73 \( ( 1 + 18 T + 306 T^{2} + 2681 T^{3} + 306 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 + 30 T + 360 T^{2} + 2028 T^{3} + 5490 T^{4} + 36318 T^{5} + 477341 T^{6} + 36318 p T^{7} + 5490 p^{2} T^{8} + 2028 p^{3} T^{9} + 360 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 6 T - 12 T^{2} - 719 T^{3} - 5007 T^{4} + 73899 T^{5} + 904205 T^{6} + 73899 p T^{7} - 5007 p^{2} T^{8} - 719 p^{3} T^{9} - 12 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 33 T + 516 T^{2} + 4394 T^{3} + 6204 T^{4} - 396549 T^{5} - 5838061 T^{6} - 396549 p T^{7} + 6204 p^{2} T^{8} + 4394 p^{3} T^{9} + 516 p^{4} T^{10} + 33 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 42 T + 897 T^{2} - 14982 T^{3} + 217518 T^{4} - 2621634 T^{5} + 27146693 T^{6} - 2621634 p T^{7} + 217518 p^{2} T^{8} - 14982 p^{3} T^{9} + 897 p^{4} T^{10} - 42 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.97874561678835773186386749229, −5.57321543914865912497327452807, −5.41064324065573300423027478912, −5.30509013183461905405886644286, −4.90089089907987623496107247415, −4.78900635766987468386920205847, −4.51617016129084077545411065928, −4.37237395445137497594708895455, −4.34081596553181111784851599697, −4.31168942095113708276264181230, −3.81435285114654424561994563069, −3.71101473347504690538527390213, −3.45387271138091834521201437581, −3.06589470445273630970187655927, −3.01861853196767020770029508910, −2.95834831558092379030718334155, −2.76479207262823042253124285365, −2.69361444408611436879072744180, −2.64075309007320788228538459151, −1.85548417965320765154064105789, −1.81258045178044227109318279009, −1.50589785911462737464270583775, −1.36456636823909439622687904057, −0.72096561873134558806469078042, −0.71962071489675331191767667752, 0.71962071489675331191767667752, 0.72096561873134558806469078042, 1.36456636823909439622687904057, 1.50589785911462737464270583775, 1.81258045178044227109318279009, 1.85548417965320765154064105789, 2.64075309007320788228538459151, 2.69361444408611436879072744180, 2.76479207262823042253124285365, 2.95834831558092379030718334155, 3.01861853196767020770029508910, 3.06589470445273630970187655927, 3.45387271138091834521201437581, 3.71101473347504690538527390213, 3.81435285114654424561994563069, 4.31168942095113708276264181230, 4.34081596553181111784851599697, 4.37237395445137497594708895455, 4.51617016129084077545411065928, 4.78900635766987468386920205847, 4.90089089907987623496107247415, 5.30509013183461905405886644286, 5.41064324065573300423027478912, 5.57321543914865912497327452807, 5.97874561678835773186386749229

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

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