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Properties

Label 12-3e36-1.1-c1e6-0-11
Degree $12$
Conductor $1.501\times 10^{17}$
Sign $1$
Analytic cond. $38907.0$
Root an. cond. $2.41269$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·2-s + 18·4-s − 3·5-s − 9·7-s − 36·8-s + 18·10-s − 15·11-s − 9·13-s + 54·14-s + 54·16-s + 9·17-s + 3·19-s − 54·20-s + 90·22-s − 12·23-s − 9·25-s + 54·26-s − 162·28-s + 3·29-s − 9·31-s − 69·32-s − 54·34-s + 27·35-s + 3·37-s − 18·38-s + 108·40-s − 21·41-s + ⋯
L(s)  = 1  − 4.24·2-s + 9·4-s − 1.34·5-s − 3.40·7-s − 12.7·8-s + 5.69·10-s − 4.52·11-s − 2.49·13-s + 14.4·14-s + 27/2·16-s + 2.18·17-s + 0.688·19-s − 12.0·20-s + 19.1·22-s − 2.50·23-s − 9/5·25-s + 10.5·26-s − 30.6·28-s + 0.557·29-s − 1.61·31-s − 12.1·32-s − 9.26·34-s + 4.56·35-s + 0.493·37-s − 2.91·38-s + 17.0·40-s − 3.27·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{36}\)
Sign: $1$
Analytic conductor: \(38907.0\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 3^{36} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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 \( 1 \)
good2 \( 1 + 3 p T + 9 p T^{2} + 9 p^{2} T^{3} + 27 p T^{4} + 69 T^{5} + 91 T^{6} + 69 p T^{7} + 27 p^{3} T^{8} + 9 p^{5} T^{9} + 9 p^{5} T^{10} + 3 p^{6} T^{11} + p^{6} T^{12} \)
5 \( 1 + 3 T + 18 T^{2} + 54 T^{3} + 189 T^{4} + 453 T^{5} + 1189 T^{6} + 453 p T^{7} + 189 p^{2} T^{8} + 54 p^{3} T^{9} + 18 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 9 T + 45 T^{2} + 173 T^{3} + 594 T^{4} + 1782 T^{5} + 4881 T^{6} + 1782 p T^{7} + 594 p^{2} T^{8} + 173 p^{3} T^{9} + 45 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 15 T + 9 p T^{2} + 333 T^{3} + 162 T^{4} - 4098 T^{5} - 21023 T^{6} - 4098 p T^{7} + 162 p^{2} T^{8} + 333 p^{3} T^{9} + 9 p^{5} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 9 T + 45 T^{2} + 191 T^{3} + 810 T^{4} + 2916 T^{5} + 10065 T^{6} + 2916 p T^{7} + 810 p^{2} T^{8} + 191 p^{3} T^{9} + 45 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
17 \( ( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{3} \)
19 \( 1 - 3 T - 24 T^{2} + 131 T^{3} + 117 T^{4} - 1116 T^{5} + 3003 T^{6} - 1116 p T^{7} + 117 p^{2} T^{8} + 131 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 12 T + 99 T^{2} + 675 T^{3} + 4455 T^{4} + 24051 T^{5} + 121258 T^{6} + 24051 p T^{7} + 4455 p^{2} T^{8} + 675 p^{3} T^{9} + 99 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 3 T - 36 T^{2} + 270 T^{3} - 945 T^{4} - 5637 T^{5} + 70993 T^{6} - 5637 p T^{7} - 945 p^{2} T^{8} + 270 p^{3} T^{9} - 36 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 9 T + 72 T^{2} + 650 T^{3} + 4293 T^{4} + 24921 T^{5} + 159267 T^{6} + 24921 p T^{7} + 4293 p^{2} T^{8} + 650 p^{3} T^{9} + 72 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 3 T - 78 T^{2} + 5 p T^{3} + 3681 T^{4} - 4194 T^{5} - 141339 T^{6} - 4194 p T^{7} + 3681 p^{2} T^{8} + 5 p^{4} T^{9} - 78 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 21 T + 207 T^{2} + 1089 T^{3} + 378 T^{4} - 52980 T^{5} - 510623 T^{6} - 52980 p T^{7} + 378 p^{2} T^{8} + 1089 p^{3} T^{9} + 207 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 9 T + 72 T^{2} + 794 T^{3} + 5697 T^{4} + 34965 T^{5} + 270831 T^{6} + 34965 p T^{7} + 5697 p^{2} T^{8} + 794 p^{3} T^{9} + 72 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 15 T + 180 T^{2} + 1476 T^{3} + 10665 T^{4} + 64275 T^{5} + 398503 T^{6} + 64275 p T^{7} + 10665 p^{2} T^{8} + 1476 p^{3} T^{9} + 180 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 + 18 T + 240 T^{2} + 1989 T^{3} + 240 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 3 T - 36 T^{2} - 972 T^{3} - 3402 T^{4} + 23781 T^{5} + 658585 T^{6} + 23781 p T^{7} - 3402 p^{2} T^{8} - 972 p^{3} T^{9} - 36 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 27 T + 270 T^{2} + 794 T^{3} - 6183 T^{4} - 65691 T^{5} - 407355 T^{6} - 65691 p T^{7} - 6183 p^{2} T^{8} + 794 p^{3} T^{9} + 270 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 27 T + 324 T^{2} - 2320 T^{3} + 4941 T^{4} + 121797 T^{5} - 1636989 T^{6} + 121797 p T^{7} + 4941 p^{2} T^{8} - 2320 p^{3} T^{9} + 324 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 9 T + 30 T^{2} - 99 T^{3} - 3531 T^{4} - 5580 T^{5} + 200671 T^{6} - 5580 p T^{7} - 3531 p^{2} T^{8} - 99 p^{3} T^{9} + 30 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 6 T - 114 T^{2} - 58 T^{3} + 8676 T^{4} - 27576 T^{5} - 826869 T^{6} - 27576 p T^{7} + 8676 p^{2} T^{8} - 58 p^{3} T^{9} - 114 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 18 T + 153 T^{2} - 853 T^{3} - 5697 T^{4} + 179145 T^{5} - 1911282 T^{6} + 179145 p T^{7} - 5697 p^{2} T^{8} - 853 p^{3} T^{9} + 153 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 15 T + 72 T^{2} - 360 T^{3} - 3159 T^{4} + 68631 T^{5} + 989803 T^{6} + 68631 p T^{7} - 3159 p^{2} T^{8} - 360 p^{3} T^{9} + 72 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 78 T^{2} - 1998 T^{3} - 858 T^{4} + 77922 T^{5} + 1866463 T^{6} + 77922 p T^{7} - 858 p^{2} T^{8} - 1998 p^{3} T^{9} - 78 p^{4} T^{10} + p^{6} T^{12} \)
97 \( 1 + 36 T + 396 T^{2} - 610 T^{3} - 33480 T^{4} + 134136 T^{5} + 5366595 T^{6} + 134136 p T^{7} - 33480 p^{2} T^{8} - 610 p^{3} T^{9} + 396 p^{4} T^{10} + 36 p^{5} T^{11} + p^{6} T^{12} \)
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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

−6.26884842238746387175008116557, −5.88373849714584139387447137164, −5.74897436891187745832791951118, −5.72036925696362439667627769453, −5.38815390458204314634861930472, −5.18012185744780377033439776125, −5.12345822805052969623259889195, −5.07487139791385402105062047126, −4.99305168930181618231918895683, −4.47179218328817879593997834614, −4.35801853833068107226980476221, −3.92262555841052825930223146402, −3.89053468481257972549392596174, −3.50678411776986203598025252829, −3.36772107857940486959429754203, −3.24701946461581273194775912274, −3.11808074763030258520557945789, −2.92882832440759430249519313068, −2.81149353380897269419651915773, −2.51527548959316819569282873996, −2.27231113389203457391997785840, −2.07119815925431077216697582968, −1.72180626619681689135093834528, −1.44668256720594348858531417484, −1.32301700358388038993176109676, 0, 0, 0, 0, 0, 0, 1.32301700358388038993176109676, 1.44668256720594348858531417484, 1.72180626619681689135093834528, 2.07119815925431077216697582968, 2.27231113389203457391997785840, 2.51527548959316819569282873996, 2.81149353380897269419651915773, 2.92882832440759430249519313068, 3.11808074763030258520557945789, 3.24701946461581273194775912274, 3.36772107857940486959429754203, 3.50678411776986203598025252829, 3.89053468481257972549392596174, 3.92262555841052825930223146402, 4.35801853833068107226980476221, 4.47179218328817879593997834614, 4.99305168930181618231918895683, 5.07487139791385402105062047126, 5.12345822805052969623259889195, 5.18012185744780377033439776125, 5.38815390458204314634861930472, 5.72036925696362439667627769453, 5.74897436891187745832791951118, 5.88373849714584139387447137164, 6.26884842238746387175008116557

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

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