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Properties

Label 12-15e12-1.1-c1e6-0-0
Degree $12$
Conductor $1.297\times 10^{14}$
Sign $1$
Analytic cond. $33.6323$
Root an. cond. $1.34038$
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  + 2-s − 3-s + 4-s − 6-s + 5·7-s − 3·9-s + 2·11-s − 12-s + 4·13-s + 5·14-s + 16-s + 4·17-s − 3·18-s + 8·19-s − 5·21-s + 2·22-s + 3·23-s + 4·26-s + 4·27-s + 5·28-s + 7·29-s − 8·31-s + 3·32-s − 2·33-s + 4·34-s − 3·36-s − 12·37-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.88·7-s − 9-s + 0.603·11-s − 0.288·12-s + 1.10·13-s + 1.33·14-s + 1/4·16-s + 0.970·17-s − 0.707·18-s + 1.83·19-s − 1.09·21-s + 0.426·22-s + 0.625·23-s + 0.784·26-s + 0.769·27-s + 0.944·28-s + 1.29·29-s − 1.43·31-s + 0.530·32-s − 0.348·33-s + 0.685·34-s − 1/2·36-s − 1.97·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 5^{12}\)
Sign: $1$
Analytic conductor: \(33.6323\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 5^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.115542064\)
\(L(\frac12)\) \(\approx\) \(4.115542064\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T + 4 T^{2} + p T^{3} + 4 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
5 \( 1 \)
good2 \( 1 - T + T^{3} - p T^{4} - T^{5} + 11 T^{6} - p T^{7} - p^{3} T^{8} + p^{3} T^{9} - p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 5 T + T^{2} + 2 p T^{3} + 73 T^{4} - 173 T^{5} - 82 T^{6} - 173 p T^{7} + 73 p^{2} T^{8} + 2 p^{4} T^{9} + p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 2 T - 21 T^{2} + 14 T^{3} + 26 p T^{4} + 58 T^{5} - 3673 T^{6} + 58 p T^{7} + 26 p^{3} T^{8} + 14 p^{3} T^{9} - 21 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 4 T - 19 T^{2} + 60 T^{3} + 370 T^{4} - 536 T^{5} - 4235 T^{6} - 536 p T^{7} + 370 p^{2} T^{8} + 60 p^{3} T^{9} - 19 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
17 \( ( 1 - 2 T + 43 T^{2} - 56 T^{3} + 43 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( ( 1 - 4 T + 53 T^{2} - 148 T^{3} + 53 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 3 T - 27 T^{2} - 66 T^{3} + 405 T^{4} + 2625 T^{5} - 9794 T^{6} + 2625 p T^{7} + 405 p^{2} T^{8} - 66 p^{3} T^{9} - 27 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 7 T - 9 T^{2} + 304 T^{3} - 803 T^{4} - 101 p T^{5} + 36038 T^{6} - 101 p^{2} T^{7} - 803 p^{2} T^{8} + 304 p^{3} T^{9} - 9 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 8 T + p T^{2} + 208 T^{3} - 158 T^{4} - 7756 T^{5} - 34897 T^{6} - 7756 p T^{7} - 158 p^{2} T^{8} + 208 p^{3} T^{9} + p^{5} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 6 T + 99 T^{2} + 440 T^{3} + 99 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 - 13 T + 27 T^{2} + 292 T^{3} + 445 T^{4} - 22279 T^{5} + 169790 T^{6} - 22279 p T^{7} + 445 p^{2} T^{8} + 292 p^{3} T^{9} + 27 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 10 T - 25 T^{2} + 462 T^{3} + 1690 T^{4} - 17990 T^{5} + 34975 T^{6} - 17990 p T^{7} + 1690 p^{2} T^{8} + 462 p^{3} T^{9} - 25 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 13 T + 39 T^{2} + 16 T^{3} - 299 T^{4} + 19253 T^{5} - 232366 T^{6} + 19253 p T^{7} - 299 p^{2} T^{8} + 16 p^{3} T^{9} + 39 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
53 \( ( 1 - 2 T + 139 T^{2} - 188 T^{3} + 139 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 2 T - 153 T^{2} + 110 T^{3} + 14962 T^{4} - 3194 T^{5} - 1012513 T^{6} - 3194 p T^{7} + 14962 p^{2} T^{8} + 110 p^{3} T^{9} - 153 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + T - 145 T^{2} - 240 T^{3} + 12217 T^{4} + 14087 T^{5} - 812786 T^{6} + 14087 p T^{7} + 12217 p^{2} T^{8} - 240 p^{3} T^{9} - 145 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 11 T - 41 T^{2} + 152 T^{3} + 4753 T^{4} + 32755 T^{5} - 764902 T^{6} + 32755 p T^{7} + 4753 p^{2} T^{8} + 152 p^{3} T^{9} - 41 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 10 T + 121 T^{2} + 712 T^{3} + 121 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 - 8 T + 155 T^{2} - 1040 T^{3} + 155 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 + 2 T - 149 T^{2} - 374 T^{3} + 10642 T^{4} + 16154 T^{5} - 772597 T^{6} + 16154 p T^{7} + 10642 p^{2} T^{8} - 374 p^{3} T^{9} - 149 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 15 T - 51 T^{2} - 678 T^{3} + 17133 T^{4} + 80979 T^{5} - 925778 T^{6} + 80979 p T^{7} + 17133 p^{2} T^{8} - 678 p^{3} T^{9} - 51 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
89 \( ( 1 + 3 T + p T^{2} )^{6} \)
97 \( 1 + 18 T + 69 T^{2} + 214 T^{3} + 906 T^{4} - 163158 T^{5} - 2743251 T^{6} - 163158 p T^{7} + 906 p^{2} T^{8} + 214 p^{3} T^{9} + 69 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
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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

−6.74085300548243412512766178579, −6.58815448682426656577357685874, −6.23104711568237990411690890343, −6.06699283303306600801057647309, −5.78242183872124389711278965647, −5.77441286458787245560042476342, −5.47337299058024000497197791779, −5.37437880269884031848871603276, −5.31282089865874750191505761249, −5.30449963151017578807392288001, −4.62207387140941616758771022501, −4.56752198857636589387562035341, −4.46184693724550207779768763123, −3.98879647062474225702994015863, −3.96750959348308853830518747073, −3.66104550217236622403085346314, −3.53067851786585736663135382453, −2.98748582797083211699318964800, −2.95859843292586796614080942386, −2.61532573891810422964373274961, −2.41409767253124081524929088249, −2.00878147987189130033841711536, −1.32047138051633242702849074123, −1.18493927446267079811972016302, −1.06023396917482204954362850798, 1.06023396917482204954362850798, 1.18493927446267079811972016302, 1.32047138051633242702849074123, 2.00878147987189130033841711536, 2.41409767253124081524929088249, 2.61532573891810422964373274961, 2.95859843292586796614080942386, 2.98748582797083211699318964800, 3.53067851786585736663135382453, 3.66104550217236622403085346314, 3.96750959348308853830518747073, 3.98879647062474225702994015863, 4.46184693724550207779768763123, 4.56752198857636589387562035341, 4.62207387140941616758771022501, 5.30449963151017578807392288001, 5.31282089865874750191505761249, 5.37437880269884031848871603276, 5.47337299058024000497197791779, 5.77441286458787245560042476342, 5.78242183872124389711278965647, 6.06699283303306600801057647309, 6.23104711568237990411690890343, 6.58815448682426656577357685874, 6.74085300548243412512766178579

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

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