Properties

Label 12-4030e6-1.1-c1e6-0-2
Degree $12$
Conductor $4.284\times 10^{21}$
Sign $1$
Analytic cond. $1.11043\times 10^{9}$
Root an. cond. $5.67271$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s − 7·3-s + 21·4-s + 6·5-s − 42·6-s − 8·7-s + 56·8-s + 19·9-s + 36·10-s − 10·11-s − 147·12-s + 6·13-s − 48·14-s − 42·15-s + 126·16-s − 14·17-s + 114·18-s − 3·19-s + 126·20-s + 56·21-s − 60·22-s − 13·23-s − 392·24-s + 21·25-s + 36·26-s − 20·27-s − 168·28-s + ⋯
L(s)  = 1  + 4.24·2-s − 4.04·3-s + 21/2·4-s + 2.68·5-s − 17.1·6-s − 3.02·7-s + 19.7·8-s + 19/3·9-s + 11.3·10-s − 3.01·11-s − 42.4·12-s + 1.66·13-s − 12.8·14-s − 10.8·15-s + 63/2·16-s − 3.39·17-s + 26.8·18-s − 0.688·19-s + 28.1·20-s + 12.2·21-s − 12.7·22-s − 2.71·23-s − 80.0·24-s + 21/5·25-s + 7.06·26-s − 3.84·27-s − 31.7·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 13^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 13^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 5^{6} \cdot 13^{6} \cdot 31^{6}\)
Sign: $1$
Analytic conductor: \(1.11043\times 10^{9}\)
Root analytic conductor: \(5.67271\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{6} \cdot 5^{6} \cdot 13^{6} \cdot 31^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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)$
bad2 \( ( 1 - T )^{6} \)
5 \( ( 1 - T )^{6} \)
13 \( ( 1 - T )^{6} \)
31 \( ( 1 - T )^{6} \)
good3 \( 1 + 7 T + 10 p T^{2} + 97 T^{3} + 85 p T^{4} + 562 T^{5} + 13 p^{4} T^{6} + 562 p T^{7} + 85 p^{3} T^{8} + 97 p^{3} T^{9} + 10 p^{5} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 8 T + 44 T^{2} + 26 p T^{3} + 653 T^{4} + 2017 T^{5} + 5615 T^{6} + 2017 p T^{7} + 653 p^{2} T^{8} + 26 p^{4} T^{9} + 44 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 10 T + 69 T^{2} + 356 T^{3} + 1543 T^{4} + 5840 T^{5} + 19901 T^{6} + 5840 p T^{7} + 1543 p^{2} T^{8} + 356 p^{3} T^{9} + 69 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 14 T + 117 T^{2} + 778 T^{3} + 4526 T^{4} + 22785 T^{5} + 100085 T^{6} + 22785 p T^{7} + 4526 p^{2} T^{8} + 778 p^{3} T^{9} + 117 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 3 T + 5 p T^{2} + 255 T^{3} + 4078 T^{4} + 9104 T^{5} + 100091 T^{6} + 9104 p T^{7} + 4078 p^{2} T^{8} + 255 p^{3} T^{9} + 5 p^{5} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 13 T + 124 T^{2} + 633 T^{3} + 2586 T^{4} + 183 p T^{5} + 623 p T^{6} + 183 p^{2} T^{7} + 2586 p^{2} T^{8} + 633 p^{3} T^{9} + 124 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 8 T + 149 T^{2} + 932 T^{3} + 9556 T^{4} + 48067 T^{5} + 353067 T^{6} + 48067 p T^{7} + 9556 p^{2} T^{8} + 932 p^{3} T^{9} + 149 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 16 T + 159 T^{2} + 852 T^{3} + 939 T^{4} - 30682 T^{5} - 270217 T^{6} - 30682 p T^{7} + 939 p^{2} T^{8} + 852 p^{3} T^{9} + 159 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 10 T + 93 T^{2} + 855 T^{3} + 6299 T^{4} + 48386 T^{5} + 346331 T^{6} + 48386 p T^{7} + 6299 p^{2} T^{8} + 855 p^{3} T^{9} + 93 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 17 T + 249 T^{2} + 2409 T^{3} + 21666 T^{4} + 158008 T^{5} + 1121063 T^{6} + 158008 p T^{7} + 21666 p^{2} T^{8} + 2409 p^{3} T^{9} + 249 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 14 T + 240 T^{2} + 2166 T^{3} + 22925 T^{4} + 159319 T^{5} + 1312487 T^{6} + 159319 p T^{7} + 22925 p^{2} T^{8} + 2166 p^{3} T^{9} + 240 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 22 T + 357 T^{2} + 4139 T^{3} + 42093 T^{4} + 358184 T^{5} + 2803597 T^{6} + 358184 p T^{7} + 42093 p^{2} T^{8} + 4139 p^{3} T^{9} + 357 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 23 T + 510 T^{2} + 6991 T^{3} + 88284 T^{4} + 835701 T^{5} + 7277105 T^{6} + 835701 p T^{7} + 88284 p^{2} T^{8} + 6991 p^{3} T^{9} + 510 p^{4} T^{10} + 23 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 17 T + 379 T^{2} - 3851 T^{3} + 49235 T^{4} - 363201 T^{5} + 3620303 T^{6} - 363201 p T^{7} + 49235 p^{2} T^{8} - 3851 p^{3} T^{9} + 379 p^{4} T^{10} - 17 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 16 T + 394 T^{2} + 4508 T^{3} + 64143 T^{4} + 556699 T^{5} + 5663575 T^{6} + 556699 p T^{7} + 64143 p^{2} T^{8} + 4508 p^{3} T^{9} + 394 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 2 T + 233 T^{2} - 1242 T^{3} + 26218 T^{4} - 190114 T^{5} + 2114785 T^{6} - 190114 p T^{7} + 26218 p^{2} T^{8} - 1242 p^{3} T^{9} + 233 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 7 T + 385 T^{2} + 2249 T^{3} + 65457 T^{4} + 309103 T^{5} + 6224551 T^{6} + 309103 p T^{7} + 65457 p^{2} T^{8} + 2249 p^{3} T^{9} + 385 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 8 T + 337 T^{2} + 2368 T^{3} + 54689 T^{4} + 332240 T^{5} + 5411813 T^{6} + 332240 p T^{7} + 54689 p^{2} T^{8} + 2368 p^{3} T^{9} + 337 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 16 T + 128 T^{2} + 284 T^{3} + 5539 T^{4} + 94179 T^{5} + 1312471 T^{6} + 94179 p T^{7} + 5539 p^{2} T^{8} + 284 p^{3} T^{9} + 128 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 32 T + 707 T^{2} + 12242 T^{3} + 172764 T^{4} + 2045741 T^{5} + 20948837 T^{6} + 2045741 p T^{7} + 172764 p^{2} T^{8} + 12242 p^{3} T^{9} + 707 p^{4} T^{10} + 32 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 31 T + 762 T^{2} - 13312 T^{3} + 195774 T^{4} - 2394801 T^{5} + 25480035 T^{6} - 2394801 p T^{7} + 195774 p^{2} T^{8} - 13312 p^{3} T^{9} + 762 p^{4} T^{10} - 31 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.96588941882004924479894413051, −4.67087761224666229352941531421, −4.64338491271236818082708040051, −4.46533155092225306713593071262, −4.35690281870961021747287546860, −4.27958323590812254683804057124, −3.96977203221822252032054884615, −3.81383936454106582223647305312, −3.55898815326130467372896312405, −3.45510873869116673100520091864, −3.42133908490650427875905350407, −3.29303826032405846163772610993, −3.27860747218273468854131463134, −2.73081815356861052945722307815, −2.70872242142060449493998993937, −2.64870957228651598177273917966, −2.57395101476605600291467459073, −2.56971161428728053575873338889, −2.16636090200251347684431665594, −1.90350291735546057381527400828, −1.78796552991875759033397939702, −1.63787768964363617144269538442, −1.57591016972136625545720678030, −1.35106114208135389384667132155, −1.19435272788870820145441889728, 0, 0, 0, 0, 0, 0, 1.19435272788870820145441889728, 1.35106114208135389384667132155, 1.57591016972136625545720678030, 1.63787768964363617144269538442, 1.78796552991875759033397939702, 1.90350291735546057381527400828, 2.16636090200251347684431665594, 2.56971161428728053575873338889, 2.57395101476605600291467459073, 2.64870957228651598177273917966, 2.70872242142060449493998993937, 2.73081815356861052945722307815, 3.27860747218273468854131463134, 3.29303826032405846163772610993, 3.42133908490650427875905350407, 3.45510873869116673100520091864, 3.55898815326130467372896312405, 3.81383936454106582223647305312, 3.96977203221822252032054884615, 4.27958323590812254683804057124, 4.35690281870961021747287546860, 4.46533155092225306713593071262, 4.64338491271236818082708040051, 4.67087761224666229352941531421, 4.96588941882004924479894413051

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

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