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Properties

Label 12-3700e6-1.1-c1e6-0-1
Degree $12$
Conductor $2.566\times 10^{21}$
Sign $1$
Analytic cond. $6.65078\times 10^{8}$
Root an. cond. $5.43549$
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-s − 2·7-s − 4·9-s − 8·11-s + 5·13-s + 2·17-s + 2·19-s − 2·21-s − 8·23-s − 5·27-s + 13·29-s − 3·31-s − 8·33-s + 6·37-s + 5·39-s + 22·41-s − 8·43-s + 18·47-s − 3·49-s + 2·51-s + 3·53-s + 2·57-s + 13·59-s − 9·61-s + 8·63-s + 27·67-s − 8·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.755·7-s − 4/3·9-s − 2.41·11-s + 1.38·13-s + 0.485·17-s + 0.458·19-s − 0.436·21-s − 1.66·23-s − 0.962·27-s + 2.41·29-s − 0.538·31-s − 1.39·33-s + 0.986·37-s + 0.800·39-s + 3.43·41-s − 1.21·43-s + 2.62·47-s − 3/7·49-s + 0.280·51-s + 0.412·53-s + 0.264·57-s + 1.69·59-s − 1.15·61-s + 1.00·63-s + 3.29·67-s − 0.963·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 5^{12} \cdot 37^{6}\)
Sign: $1$
Analytic conductor: \(6.65078\times 10^{8}\)
Root analytic conductor: \(5.43549\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 5^{12} \cdot 37^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.941167394\)
\(L(\frac12)\) \(\approx\) \(2.941167394\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( ( 1 - T )^{6} \)
good3 \( 1 - T + 5 T^{2} - 4 T^{3} + 11 T^{4} + 8 T^{5} + 28 T^{6} + 8 p T^{7} + 11 p^{2} T^{8} - 4 p^{3} T^{9} + 5 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 2 T + p T^{2} + 4 T^{3} + p^{2} T^{4} - 55 T^{5} - 88 T^{6} - 55 p T^{7} + p^{4} T^{8} + 4 p^{3} T^{9} + p^{5} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 8 T + 75 T^{2} + 397 T^{3} + 2170 T^{4} + 8354 T^{5} + 32234 T^{6} + 8354 p T^{7} + 2170 p^{2} T^{8} + 397 p^{3} T^{9} + 75 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 5 T + 41 T^{2} - 150 T^{3} + 733 T^{4} - 2254 T^{5} + 9166 T^{6} - 2254 p T^{7} + 733 p^{2} T^{8} - 150 p^{3} T^{9} + 41 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 2 T + 54 T^{2} - 7 T^{3} + 1159 T^{4} + 2116 T^{5} + 17732 T^{6} + 2116 p T^{7} + 1159 p^{2} T^{8} - 7 p^{3} T^{9} + 54 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 2 T + 50 T^{2} - 78 T^{3} + 892 T^{4} - 1513 T^{5} + 11641 T^{6} - 1513 p T^{7} + 892 p^{2} T^{8} - 78 p^{3} T^{9} + 50 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 8 T + 102 T^{2} + 619 T^{3} + 4990 T^{4} + 23915 T^{5} + 144401 T^{6} + 23915 p T^{7} + 4990 p^{2} T^{8} + 619 p^{3} T^{9} + 102 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 13 T + 231 T^{2} - 1970 T^{3} + 19201 T^{4} - 116578 T^{5} + 773822 T^{6} - 116578 p T^{7} + 19201 p^{2} T^{8} - 1970 p^{3} T^{9} + 231 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 3 T + 117 T^{2} + 11 p T^{3} + 7410 T^{4} + 18489 T^{5} + 283104 T^{6} + 18489 p T^{7} + 7410 p^{2} T^{8} + 11 p^{4} T^{9} + 117 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 22 T + 297 T^{2} - 2903 T^{3} + 23095 T^{4} - 160117 T^{5} + 1056149 T^{6} - 160117 p T^{7} + 23095 p^{2} T^{8} - 2903 p^{3} T^{9} + 297 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 8 T + 214 T^{2} + 1483 T^{3} + 20674 T^{4} + 118391 T^{5} + 1144049 T^{6} + 118391 p T^{7} + 20674 p^{2} T^{8} + 1483 p^{3} T^{9} + 214 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 18 T + 315 T^{2} - 3330 T^{3} + 34137 T^{4} - 264699 T^{5} + 43244 p T^{6} - 264699 p T^{7} + 34137 p^{2} T^{8} - 3330 p^{3} T^{9} + 315 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 3 T + 291 T^{2} - 723 T^{3} + 36555 T^{4} - 73092 T^{5} + 2537827 T^{6} - 73092 p T^{7} + 36555 p^{2} T^{8} - 723 p^{3} T^{9} + 291 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 13 T + 285 T^{2} - 2768 T^{3} + 34591 T^{4} - 269662 T^{5} + 2506175 T^{6} - 269662 p T^{7} + 34591 p^{2} T^{8} - 2768 p^{3} T^{9} + 285 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 9 T + 327 T^{2} + 2357 T^{3} + 46938 T^{4} + 268131 T^{5} + 3739518 T^{6} + 268131 p T^{7} + 46938 p^{2} T^{8} + 2357 p^{3} T^{9} + 327 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 27 T + 546 T^{2} - 7000 T^{3} + 78414 T^{4} - 691140 T^{5} + 6075138 T^{6} - 691140 p T^{7} + 78414 p^{2} T^{8} - 7000 p^{3} T^{9} + 546 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 20 T + 375 T^{2} - 4858 T^{3} + 60301 T^{4} - 582275 T^{5} + 5408696 T^{6} - 582275 p T^{7} + 60301 p^{2} T^{8} - 4858 p^{3} T^{9} + 375 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 41 T + 1066 T^{2} + 19525 T^{3} + 279127 T^{4} + 43898 p T^{5} + 30168860 T^{6} + 43898 p^{2} T^{7} + 279127 p^{2} T^{8} + 19525 p^{3} T^{9} + 1066 p^{4} T^{10} + 41 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 14 T + 263 T^{2} - 2721 T^{3} + 34729 T^{4} - 313357 T^{5} + 3298411 T^{6} - 313357 p T^{7} + 34729 p^{2} T^{8} - 2721 p^{3} T^{9} + 263 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 16 T + 474 T^{2} + 6089 T^{3} + 95503 T^{4} + 969406 T^{5} + 10467596 T^{6} + 969406 p T^{7} + 95503 p^{2} T^{8} + 6089 p^{3} T^{9} + 474 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 16 T + 318 T^{2} - 3137 T^{3} + 40393 T^{4} - 311974 T^{5} + 3692852 T^{6} - 311974 p T^{7} + 40393 p^{2} T^{8} - 3137 p^{3} T^{9} + 318 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 23 T + 545 T^{2} - 7935 T^{3} + 116746 T^{4} - 1299493 T^{5} + 14500966 T^{6} - 1299493 p T^{7} + 116746 p^{2} T^{8} - 7935 p^{3} T^{9} + 545 p^{4} T^{10} - 23 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.36006561154972952985824510847, −4.05863230173712020142837137967, −4.04328555550941398105379186042, −4.00365261900621758955357015118, −3.72372084879060759450076524345, −3.68497393714896891651337388640, −3.59304558596748366760480362583, −3.19956985999144782331156224249, −3.19075227090506269613992723792, −3.02025766005956360444640412219, −2.97366880805371520965114153171, −2.78492954425292238858768616455, −2.60069513334318807597201579469, −2.40131111295735457355070289743, −2.29715082655407288063543078143, −2.24488901249233124721411474071, −2.09822105498877571150462415941, −1.87787449843072093905790885097, −1.63587124350493114916181870853, −1.24543592725286629091677352609, −0.914855393448397622857874879754, −0.865947272315375997931795314031, −0.820708798916795464513596911880, −0.49880379218982026886730112473, −0.17717692241753600759154278714, 0.17717692241753600759154278714, 0.49880379218982026886730112473, 0.820708798916795464513596911880, 0.865947272315375997931795314031, 0.914855393448397622857874879754, 1.24543592725286629091677352609, 1.63587124350493114916181870853, 1.87787449843072093905790885097, 2.09822105498877571150462415941, 2.24488901249233124721411474071, 2.29715082655407288063543078143, 2.40131111295735457355070289743, 2.60069513334318807597201579469, 2.78492954425292238858768616455, 2.97366880805371520965114153171, 3.02025766005956360444640412219, 3.19075227090506269613992723792, 3.19956985999144782331156224249, 3.59304558596748366760480362583, 3.68497393714896891651337388640, 3.72372084879060759450076524345, 4.00365261900621758955357015118, 4.04328555550941398105379186042, 4.05863230173712020142837137967, 4.36006561154972952985824510847

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

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