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Properties

Label 12-930e6-1.1-c1e6-0-0
Degree $12$
Conductor $6.470\times 10^{17}$
Sign $1$
Analytic cond. $167710.$
Root an. cond. $2.72508$
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  − 6·2-s + 3·3-s + 21·4-s − 3·5-s − 18·6-s − 4·7-s − 56·8-s + 3·9-s + 18·10-s + 5·11-s + 63·12-s − 2·13-s + 24·14-s − 9·15-s + 126·16-s − 11·17-s − 18·18-s − 2·19-s − 63·20-s − 12·21-s − 30·22-s + 14·23-s − 168·24-s + 3·25-s + 12·26-s − 2·27-s − 84·28-s + ⋯
L(s)  = 1  − 4.24·2-s + 1.73·3-s + 21/2·4-s − 1.34·5-s − 7.34·6-s − 1.51·7-s − 19.7·8-s + 9-s + 5.69·10-s + 1.50·11-s + 18.1·12-s − 0.554·13-s + 6.41·14-s − 2.32·15-s + 63/2·16-s − 2.66·17-s − 4.24·18-s − 0.458·19-s − 14.0·20-s − 2.61·21-s − 6.39·22-s + 2.91·23-s − 34.2·24-s + 3/5·25-s + 2.35·26-s − 0.384·27-s − 15.8·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 31^{6}\)
Sign: $1$
Analytic conductor: \(167710.\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 31^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.01170627537\)
\(L(\frac12)\) \(\approx\) \(0.01170627537\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T )^{6} \)
3 \( ( 1 - T + T^{2} )^{3} \)
5 \( ( 1 + T + T^{2} )^{3} \)
31 \( 1 - 8 T - 28 T^{2} + 508 T^{3} - 28 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
good7 \( 1 + 4 T - 24 T^{3} - 32 T^{4} + 4 T^{5} + 46 T^{6} + 4 p T^{7} - 32 p^{2} T^{8} - 24 p^{3} T^{9} + 4 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 5 T + 21 T^{2} - 58 T^{3} + 67 T^{4} - 89 T^{5} - 178 T^{6} - 89 p T^{7} + 67 p^{2} T^{8} - 58 p^{3} T^{9} + 21 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 2 T + 5 T^{2} - 42 T^{3} - 2 p T^{4} + 370 T^{5} + 4105 T^{6} + 370 p T^{7} - 2 p^{3} T^{8} - 42 p^{3} T^{9} + 5 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 11 T + 40 T^{2} + 7 p T^{3} + 938 T^{4} + 4151 T^{5} + 12316 T^{6} + 4151 p T^{7} + 938 p^{2} T^{8} + 7 p^{4} T^{9} + 40 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 2 T + 7 T^{2} + 130 T^{3} + 202 T^{4} + 1154 T^{5} + 16727 T^{6} + 1154 p T^{7} + 202 p^{2} T^{8} + 130 p^{3} T^{9} + 7 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
23 \( ( 1 - 7 T + 75 T^{2} - 310 T^{3} + 75 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( ( 1 - 12 T + 96 T^{2} - 550 T^{3} + 96 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 17 T + 92 T^{2} - 585 T^{3} + 8026 T^{4} - 47701 T^{5} + 179656 T^{6} - 47701 p T^{7} + 8026 p^{2} T^{8} - 585 p^{3} T^{9} + 92 p^{4} T^{10} - 17 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 12 T + 69 T^{2} + 228 T^{3} - 1482 T^{4} - 28428 T^{5} - 221951 T^{6} - 28428 p T^{7} - 1482 p^{2} T^{8} + 228 p^{3} T^{9} + 69 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 3 T - 84 T^{2} - 269 T^{3} + 3828 T^{4} + 7035 T^{5} - 154458 T^{6} + 7035 p T^{7} + 3828 p^{2} T^{8} - 269 p^{3} T^{9} - 84 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( ( 1 - 3 T + 51 T^{2} - 174 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( 1 - 2 T - 100 T^{2} + 32 T^{3} + 5064 T^{4} + 3950 T^{5} - 272330 T^{6} + 3950 p T^{7} + 5064 p^{2} T^{8} + 32 p^{3} T^{9} - 100 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 12 T + 12 T^{2} - 600 T^{3} - 3468 T^{4} - 996 T^{5} + 73942 T^{6} - 996 p T^{7} - 3468 p^{2} T^{8} - 600 p^{3} T^{9} + 12 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
61 \( ( 1 + 8 T + 143 T^{2} + 728 T^{3} + 143 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 + 17 T + 2 T^{2} + 75 T^{3} + 22306 T^{4} + 111901 T^{5} - 444914 T^{6} + 111901 p T^{7} + 22306 p^{2} T^{8} + 75 p^{3} T^{9} + 2 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 10 T - 105 T^{2} + 598 T^{3} + 14230 T^{4} - 30730 T^{5} - 1019077 T^{6} - 30730 p T^{7} + 14230 p^{2} T^{8} + 598 p^{3} T^{9} - 105 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
73 \( ( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{3} \)
79 \( 1 + 3 T - 192 T^{2} - 377 T^{3} + 23052 T^{4} + 21939 T^{5} - 2021670 T^{6} + 21939 p T^{7} + 23052 p^{2} T^{8} - 377 p^{3} T^{9} - 192 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 72 T^{2} - 424 T^{3} - 792 T^{4} + 15264 T^{5} + 758246 T^{6} + 15264 p T^{7} - 792 p^{2} T^{8} - 424 p^{3} T^{9} - 72 p^{4} T^{10} + p^{6} T^{12} \)
89 \( ( 1 + 28 T + 487 T^{2} + 5416 T^{3} + 487 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( ( 1 - 20 T + 172 T^{2} - 1114 T^{3} + 172 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−5.61649871989578060818825006330, −5.15107489570603082739643358572, −4.86245588915534514469288005656, −4.72662315502910275863687209632, −4.61696048569863638550608068796, −4.42077045622630873325299285961, −4.36305101590847907561901960653, −4.00727056858334276845732019987, −3.92009140314902500324853090463, −3.72151736743783107257045965508, −3.32104770820329148273444283673, −3.04990005647335337851843356199, −2.97816548262144184881540877917, −2.88299940335753496880303075994, −2.86528406495466777377553116107, −2.72037108129033903323196737598, −2.52631870974947639674155129157, −2.11664973311103435555120863408, −2.03415528666099291936647266846, −1.66261993017766728665428778203, −1.38199336531307798293077722610, −1.06979740902724276211736670392, −0.830819713454785397026578658730, −0.75966601704925162593160717689, −0.04299369751476599777834216751, 0.04299369751476599777834216751, 0.75966601704925162593160717689, 0.830819713454785397026578658730, 1.06979740902724276211736670392, 1.38199336531307798293077722610, 1.66261993017766728665428778203, 2.03415528666099291936647266846, 2.11664973311103435555120863408, 2.52631870974947639674155129157, 2.72037108129033903323196737598, 2.86528406495466777377553116107, 2.88299940335753496880303075994, 2.97816548262144184881540877917, 3.04990005647335337851843356199, 3.32104770820329148273444283673, 3.72151736743783107257045965508, 3.92009140314902500324853090463, 4.00727056858334276845732019987, 4.36305101590847907561901960653, 4.42077045622630873325299285961, 4.61696048569863638550608068796, 4.72662315502910275863687209632, 4.86245588915534514469288005656, 5.15107489570603082739643358572, 5.61649871989578060818825006330

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

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