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Properties

Label 12-378e6-1.1-c1e6-0-3
Degree $12$
Conductor $2.917\times 10^{15}$
Sign $1$
Analytic cond. $756.159$
Root an. cond. $1.73733$
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  + 3·5-s − 8-s + 6·11-s − 3·13-s + 3·19-s + 6·23-s + 9·25-s − 9·27-s − 3·29-s − 3·31-s + 12·37-s − 3·40-s − 12·41-s − 6·43-s + 24·47-s + 18·55-s − 24·59-s − 3·61-s − 9·65-s + 3·67-s − 12·71-s + 3·73-s + 33·79-s − 6·88-s + 21·89-s + 9·95-s − 15·97-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.353·8-s + 1.80·11-s − 0.832·13-s + 0.688·19-s + 1.25·23-s + 9/5·25-s − 1.73·27-s − 0.557·29-s − 0.538·31-s + 1.97·37-s − 0.474·40-s − 1.87·41-s − 0.914·43-s + 3.50·47-s + 2.42·55-s − 3.12·59-s − 0.384·61-s − 1.11·65-s + 0.366·67-s − 1.42·71-s + 0.351·73-s + 3.71·79-s − 0.639·88-s + 2.22·89-s + 0.923·95-s − 1.52·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{18} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(756.159\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{18} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.792572965\)
\(L(\frac12)\) \(\approx\) \(4.792572965\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{3} + T^{6} \)
3 \( 1 + p^{2} T^{3} + p^{3} T^{6} \)
7 \( 1 + T^{3} + T^{6} \)
good5 \( 1 - 3 T + 18 T^{3} - 36 T^{4} - 3 p^{2} T^{5} + 379 T^{6} - 3 p^{3} T^{7} - 36 p^{2} T^{8} + 18 p^{3} T^{9} - 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 6 T + 18 T^{2} - 36 T^{3} - 108 T^{4} + 912 T^{5} - 3203 T^{6} + 912 p T^{7} - 108 p^{2} T^{8} - 36 p^{3} T^{9} + 18 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 3 T + 6 T^{2} + 8 T^{3} - 27 T^{4} - 891 T^{5} - 3483 T^{6} - 891 p T^{7} - 27 p^{2} T^{8} + 8 p^{3} T^{9} + 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 42 T^{2} + 18 T^{3} + 1050 T^{4} - 378 T^{5} - 20081 T^{6} - 378 p T^{7} + 1050 p^{2} T^{8} + 18 p^{3} T^{9} - 42 p^{4} T^{10} + p^{6} T^{12} \)
19 \( 1 - 3 T - 24 T^{2} + 23 T^{3} + 279 T^{4} + 666 T^{5} - 6501 T^{6} + 666 p T^{7} + 279 p^{2} T^{8} + 23 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 6 T - 36 T^{2} + 396 T^{3} - 810 T^{4} - 5874 T^{5} + 51733 T^{6} - 5874 p T^{7} - 810 p^{2} T^{8} + 396 p^{3} T^{9} - 36 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 3 T + 108 T^{2} + 90 T^{3} + 4851 T^{4} - 3507 T^{5} + 149293 T^{6} - 3507 p T^{7} + 4851 p^{2} T^{8} + 90 p^{3} T^{9} + 108 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 3 T + 24 T^{2} + 296 T^{3} + 1530 T^{4} + 7119 T^{5} + 55665 T^{6} + 7119 p T^{7} + 1530 p^{2} T^{8} + 296 p^{3} T^{9} + 24 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 12 T + 21 T^{2} + 284 T^{3} + 18 T^{4} - 15444 T^{5} + 118797 T^{6} - 15444 p T^{7} + 18 p^{2} T^{8} + 284 p^{3} T^{9} + 21 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 12 T + 144 T^{2} + 1044 T^{3} + 9216 T^{4} + 58152 T^{5} + 440263 T^{6} + 58152 p T^{7} + 9216 p^{2} T^{8} + 1044 p^{3} T^{9} + 144 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 6 T - 6 T^{2} + 284 T^{3} - 684 T^{4} - 7920 T^{5} + 123837 T^{6} - 7920 p T^{7} - 684 p^{2} T^{8} + 284 p^{3} T^{9} - 6 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 24 T + 306 T^{2} - 2844 T^{3} + 25254 T^{4} - 215106 T^{5} + 1621333 T^{6} - 215106 p T^{7} + 25254 p^{2} T^{8} - 2844 p^{3} T^{9} + 306 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 + 42 T^{2} + 153 T^{3} + 42 p T^{4} + p^{3} T^{6} )^{2} \)
59 \( 1 + 24 T + 252 T^{2} + 1395 T^{3} - 2475 T^{4} - 150699 T^{5} - 1633823 T^{6} - 150699 p T^{7} - 2475 p^{2} T^{8} + 1395 p^{3} T^{9} + 252 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 3 T + 114 T^{2} + 926 T^{3} + 12897 T^{4} + 70029 T^{5} + 1006281 T^{6} + 70029 p T^{7} + 12897 p^{2} T^{8} + 926 p^{3} T^{9} + 114 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 3 T - 96 T^{2} + 1094 T^{3} - 3735 T^{4} - 48879 T^{5} + 940191 T^{6} - 48879 p T^{7} - 3735 p^{2} T^{8} + 1094 p^{3} T^{9} - 96 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 12 T - 24 T^{2} - 738 T^{3} - 228 T^{4} + 5556 T^{5} - 117857 T^{6} + 5556 p T^{7} - 228 p^{2} T^{8} - 738 p^{3} T^{9} - 24 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 3 T - 96 T^{2} + 635 T^{3} + 1935 T^{4} - 19872 T^{5} + 150873 T^{6} - 19872 p T^{7} + 1935 p^{2} T^{8} + 635 p^{3} T^{9} - 96 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 33 T + 492 T^{2} - 4402 T^{3} + 17532 T^{4} + 134901 T^{5} - 2452347 T^{6} + 134901 p T^{7} + 17532 p^{2} T^{8} - 4402 p^{3} T^{9} + 492 p^{4} T^{10} - 33 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 144 T^{2} + 774 T^{3} + 252 p T^{4} + 72810 T^{5} + 2086741 T^{6} + 72810 p T^{7} + 252 p^{3} T^{8} + 774 p^{3} T^{9} + 144 p^{4} T^{10} + p^{6} T^{12} \)
89 \( 1 - 21 T + 138 T^{2} + 891 T^{3} - 11325 T^{4} - 132708 T^{5} + 2900905 T^{6} - 132708 p T^{7} - 11325 p^{2} T^{8} + 891 p^{3} T^{9} + 138 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 15 T + 120 T^{2} + 2138 T^{3} + 14679 T^{4} + 127917 T^{5} + 2485869 T^{6} + 127917 p T^{7} + 14679 p^{2} T^{8} + 2138 p^{3} T^{9} + 120 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
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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

−6.19469640954300286742955724894, −5.89938642267690145183335393344, −5.88689647891834501647123125427, −5.77420506695725453505405418021, −5.33353953963057264739781476701, −5.33058048778540071897575458397, −5.25446407535737281669262475341, −4.88112144170358287409169555860, −4.48056608614308113383340474784, −4.44483479195359510960871454432, −4.42752731918502751554233371761, −4.42246261116115003567997277719, −3.63334277362969873672709770248, −3.59192359598209276680384063153, −3.51306862946926595578816879310, −3.16563681769817311945100985196, −3.00748067489457519661814878869, −2.98565728353735754904141128032, −2.25936344599167431064747569560, −2.13104643515048270328639315970, −1.99239890228082090450145361907, −1.88344327181844075184732071251, −1.36587140397128264326838243014, −0.837443845706437985596845036752, −0.75551355626911789136133263968, 0.75551355626911789136133263968, 0.837443845706437985596845036752, 1.36587140397128264326838243014, 1.88344327181844075184732071251, 1.99239890228082090450145361907, 2.13104643515048270328639315970, 2.25936344599167431064747569560, 2.98565728353735754904141128032, 3.00748067489457519661814878869, 3.16563681769817311945100985196, 3.51306862946926595578816879310, 3.59192359598209276680384063153, 3.63334277362969873672709770248, 4.42246261116115003567997277719, 4.42752731918502751554233371761, 4.44483479195359510960871454432, 4.48056608614308113383340474784, 4.88112144170358287409169555860, 5.25446407535737281669262475341, 5.33058048778540071897575458397, 5.33353953963057264739781476701, 5.77420506695725453505405418021, 5.88689647891834501647123125427, 5.89938642267690145183335393344, 6.19469640954300286742955724894

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

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