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Properties

Label 12-175e6-1.1-c1e6-0-0
Degree $12$
Conductor $2.872\times 10^{13}$
Sign $1$
Analytic cond. $7.44544$
Root an. cond. $1.18210$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3·3-s + 2·4-s − 3·6-s − 2·7-s + 3·8-s + 7·9-s + 2·11-s − 6·12-s + 8·13-s − 2·14-s + 6·16-s − 10·17-s + 7·18-s + 4·19-s + 6·21-s + 2·22-s − 23-s − 9·24-s + 8·26-s − 14·27-s − 4·28-s − 16·29-s − 5·31-s + 8·32-s − 6·33-s − 10·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 4-s − 1.22·6-s − 0.755·7-s + 1.06·8-s + 7/3·9-s + 0.603·11-s − 1.73·12-s + 2.21·13-s − 0.534·14-s + 3/2·16-s − 2.42·17-s + 1.64·18-s + 0.917·19-s + 1.30·21-s + 0.426·22-s − 0.208·23-s − 1.83·24-s + 1.56·26-s − 2.69·27-s − 0.755·28-s − 2.97·29-s − 0.898·31-s + 1.41·32-s − 1.04·33-s − 1.71·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(5^{12} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(7.44544\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 5^{12} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.621675082\)
\(L(\frac12)\) \(\approx\) \(1.621675082\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + 2 T - 4 T^{2} - 13 T^{3} - 4 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
good2 \( 1 - T - T^{2} - T^{4} + p T^{5} + 5 T^{6} + p^{2} T^{7} - p^{2} T^{8} - p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
3 \( 1 + p T + 2 T^{2} - T^{3} - 5 T^{4} - 22 T^{5} - 53 T^{6} - 22 p T^{7} - 5 p^{2} T^{8} - p^{3} T^{9} + 2 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
11 \( 1 - 2 T - 10 T^{2} - 30 T^{3} + 4 p T^{4} + 388 T^{5} + 23 T^{6} + 388 p T^{7} + 4 p^{3} T^{8} - 30 p^{3} T^{9} - 10 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 - 4 T + 24 T^{2} - 111 T^{3} + 24 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 + 10 T + 20 T^{2} + 78 T^{3} + 1430 T^{4} + 4570 T^{5} + 2447 T^{6} + 4570 p T^{7} + 1430 p^{2} T^{8} + 78 p^{3} T^{9} + 20 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 4 T - 24 T^{2} + 158 T^{3} + 204 T^{4} - 1724 T^{5} + 2823 T^{6} - 1724 p T^{7} + 204 p^{2} T^{8} + 158 p^{3} T^{9} - 24 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + T - 58 T^{2} - 51 T^{3} + 2069 T^{4} + 1060 T^{5} - 53233 T^{6} + 1060 p T^{7} + 2069 p^{2} T^{8} - 51 p^{3} T^{9} - 58 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 + 8 T + 82 T^{2} + 389 T^{3} + 82 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 + 5 T - 50 T^{2} - 119 T^{3} + 2065 T^{4} + 80 T^{5} - 77809 T^{6} + 80 p T^{7} + 2065 p^{2} T^{8} - 119 p^{3} T^{9} - 50 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 10 T - 20 T^{2} + 118 T^{3} + 3370 T^{4} - 3970 T^{5} - 134113 T^{6} - 3970 p T^{7} + 3370 p^{2} T^{8} + 118 p^{3} T^{9} - 20 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 + T + 77 T^{2} + 187 T^{3} + 77 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 - 9 T + 109 T^{2} - 721 T^{3} + 109 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 + 28 T + 390 T^{2} + 4298 T^{3} + 42442 T^{4} + 351442 T^{5} + 2516387 T^{6} + 351442 p T^{7} + 42442 p^{2} T^{8} + 4298 p^{3} T^{9} + 390 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 10 T - 40 T^{2} - 474 T^{3} + 2810 T^{4} + 10870 T^{5} - 143977 T^{6} + 10870 p T^{7} + 2810 p^{2} T^{8} - 474 p^{3} T^{9} - 40 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 5 T - 90 T^{2} + 815 T^{3} + 2965 T^{4} - 29480 T^{5} + 26483 T^{6} - 29480 p T^{7} + 2965 p^{2} T^{8} + 815 p^{3} T^{9} - 90 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 5 T - 140 T^{2} - 269 T^{3} + 14425 T^{4} + 8210 T^{5} - 1000579 T^{6} + 8210 p T^{7} + 14425 p^{2} T^{8} - 269 p^{3} T^{9} - 140 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 12 T - 85 T^{2} + 388 T^{3} + 18282 T^{4} - 44708 T^{5} - 1099813 T^{6} - 44708 p T^{7} + 18282 p^{2} T^{8} + 388 p^{3} T^{9} - 85 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 7 T + 124 T^{2} - 571 T^{3} + 124 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 12 T - 43 T^{2} + 364 T^{3} + 5670 T^{4} + 19828 T^{5} - 841663 T^{6} + 19828 p T^{7} + 5670 p^{2} T^{8} + 364 p^{3} T^{9} - 43 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 7 T - 96 T^{2} + 1499 T^{3} + 771 T^{4} - 65582 T^{5} + 599343 T^{6} - 65582 p T^{7} + 771 p^{2} T^{8} + 1499 p^{3} T^{9} - 96 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 - 26 T + 448 T^{2} - 4715 T^{3} + 448 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 6 T - 94 T^{2} - 198 T^{3} + 3584 T^{4} + 53484 T^{5} - 294637 T^{6} + 53484 p T^{7} + 3584 p^{2} T^{8} - 198 p^{3} T^{9} - 94 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 + 7 T + 181 T^{2} + 1617 T^{3} + 181 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−6.87172680040498541431639721000, −6.73495021756339601326651716516, −6.45377054014662982732249287708, −6.40900110495762122010445579983, −6.35026507561492060147430576185, −6.30447759891891551142423023054, −5.82844492260893273886817087366, −5.64840511101881924620461562944, −5.49923825304968955914079592195, −5.39529127587405852710494109408, −5.08493643213998756224425106492, −4.70777885149513025020217949877, −4.56236007397567721308183703319, −4.53821655631866208351018071442, −3.98908992622794765262544897374, −3.70884199172628340262808850905, −3.62369200916419755262741956313, −3.58814432132526316656943504851, −3.48610303891848195446308819879, −2.59070224321773225100994005398, −2.47694850271709207122483506619, −2.03222849352917142889132185740, −1.71575119030286197510240824838, −1.37701404489394181092447991407, −0.74382357131566779211374008525, 0.74382357131566779211374008525, 1.37701404489394181092447991407, 1.71575119030286197510240824838, 2.03222849352917142889132185740, 2.47694850271709207122483506619, 2.59070224321773225100994005398, 3.48610303891848195446308819879, 3.58814432132526316656943504851, 3.62369200916419755262741956313, 3.70884199172628340262808850905, 3.98908992622794765262544897374, 4.53821655631866208351018071442, 4.56236007397567721308183703319, 4.70777885149513025020217949877, 5.08493643213998756224425106492, 5.39529127587405852710494109408, 5.49923825304968955914079592195, 5.64840511101881924620461562944, 5.82844492260893273886817087366, 6.30447759891891551142423023054, 6.35026507561492060147430576185, 6.40900110495762122010445579983, 6.45377054014662982732249287708, 6.73495021756339601326651716516, 6.87172680040498541431639721000

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

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