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Properties

Label 12-4018e6-1.1-c1e6-0-1
Degree $12$
Conductor $4.208\times 10^{21}$
Sign $1$
Analytic cond. $1.09074\times 10^{9}$
Root an. cond. $5.66426$
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  − 6·2-s − 3-s + 21·4-s + 6·6-s − 56·8-s − 6·9-s + 11-s − 21·12-s − 4·13-s + 126·16-s + 17-s + 36·18-s + 3·19-s − 6·22-s + 21·23-s + 56·24-s − 15·25-s + 24·26-s + 3·27-s + 5·29-s − 3·31-s − 252·32-s − 33-s − 6·34-s − 126·36-s − 2·37-s − 18·38-s + ⋯
L(s)  = 1  − 4.24·2-s − 0.577·3-s + 21/2·4-s + 2.44·6-s − 19.7·8-s − 2·9-s + 0.301·11-s − 6.06·12-s − 1.10·13-s + 63/2·16-s + 0.242·17-s + 8.48·18-s + 0.688·19-s − 1.27·22-s + 4.37·23-s + 11.4·24-s − 3·25-s + 4.70·26-s + 0.577·27-s + 0.928·29-s − 0.538·31-s − 44.5·32-s − 0.174·33-s − 1.02·34-s − 21·36-s − 0.328·37-s − 2.91·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{12} \cdot 41^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{12} \cdot 41^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 7^{12} \cdot 41^{6}\)
Sign: $1$
Analytic conductor: \(1.09074\times 10^{9}\)
Root analytic conductor: \(5.66426\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 7^{12} \cdot 41^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5836852961\)
\(L(\frac12)\) \(\approx\) \(0.5836852961\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T )^{6} \)
7 \( 1 \)
41 \( ( 1 + T )^{6} \)
good3 \( 1 + T + 7 T^{2} + 10 T^{3} + 11 p T^{4} + 41 T^{5} + 122 T^{6} + 41 p T^{7} + 11 p^{3} T^{8} + 10 p^{3} T^{9} + 7 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 + 3 p T^{2} - T^{3} + 131 T^{4} - 3 T^{5} + 798 T^{6} - 3 p T^{7} + 131 p^{2} T^{8} - p^{3} T^{9} + 3 p^{5} T^{10} + p^{6} T^{12} \)
11 \( 1 - T + 28 T^{2} - 48 T^{3} + 463 T^{4} - 1027 T^{5} + 5388 T^{6} - 1027 p T^{7} + 463 p^{2} T^{8} - 48 p^{3} T^{9} + 28 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 4 T + 53 T^{2} + 10 p T^{3} + 1064 T^{4} + 1632 T^{5} + 14139 T^{6} + 1632 p T^{7} + 1064 p^{2} T^{8} + 10 p^{4} T^{9} + 53 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - T + 86 T^{2} - 90 T^{3} + 3291 T^{4} - 3133 T^{5} + 71976 T^{6} - 3133 p T^{7} + 3291 p^{2} T^{8} - 90 p^{3} T^{9} + 86 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 3 T + 58 T^{2} - 272 T^{3} + 2021 T^{4} - 8421 T^{5} + 49416 T^{6} - 8421 p T^{7} + 2021 p^{2} T^{8} - 272 p^{3} T^{9} + 58 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 21 T + 286 T^{2} - 2780 T^{3} + 21501 T^{4} - 135231 T^{5} + 710192 T^{6} - 135231 p T^{7} + 21501 p^{2} T^{8} - 2780 p^{3} T^{9} + 286 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 5 T + 158 T^{2} - 624 T^{3} + 10767 T^{4} - 33725 T^{5} + 407883 T^{6} - 33725 p T^{7} + 10767 p^{2} T^{8} - 624 p^{3} T^{9} + 158 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 3 T + 133 T^{2} + 238 T^{3} + 7871 T^{4} + 8247 T^{5} + 291654 T^{6} + 8247 p T^{7} + 7871 p^{2} T^{8} + 238 p^{3} T^{9} + 133 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 2 T + 81 T^{2} + 293 T^{3} + 4297 T^{4} + 15765 T^{5} + 170250 T^{6} + 15765 p T^{7} + 4297 p^{2} T^{8} + 293 p^{3} T^{9} + 81 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 12 T + 207 T^{2} - 36 p T^{3} + 15734 T^{4} - 87500 T^{5} + 747019 T^{6} - 87500 p T^{7} + 15734 p^{2} T^{8} - 36 p^{4} T^{9} + 207 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 18 T + 367 T^{2} - 4187 T^{3} + 48285 T^{4} - 390765 T^{5} + 3124970 T^{6} - 390765 p T^{7} + 48285 p^{2} T^{8} - 4187 p^{3} T^{9} + 367 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 14 T + 243 T^{2} - 2555 T^{3} + 28319 T^{4} - 233939 T^{5} + 1903710 T^{6} - 233939 p T^{7} + 28319 p^{2} T^{8} - 2555 p^{3} T^{9} + 243 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 16 T + 238 T^{2} + 2547 T^{3} + 26935 T^{4} + 227428 T^{5} + 1910883 T^{6} + 227428 p T^{7} + 26935 p^{2} T^{8} + 2547 p^{3} T^{9} + 238 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 20 T + 453 T^{2} - 5909 T^{3} + 76465 T^{4} - 709515 T^{5} + 6412098 T^{6} - 709515 p T^{7} + 76465 p^{2} T^{8} - 5909 p^{3} T^{9} + 453 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 13 T + 176 T^{2} + 1888 T^{3} + 17413 T^{4} + 153731 T^{5} + 1305512 T^{6} + 153731 p T^{7} + 17413 p^{2} T^{8} + 1888 p^{3} T^{9} + 176 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 11 T + 136 T^{2} + 162 T^{3} + 2821 T^{4} + 3931 T^{5} + 799059 T^{6} + 3931 p T^{7} + 2821 p^{2} T^{8} + 162 p^{3} T^{9} + 136 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + T + 254 T^{2} - 330 T^{3} + 30443 T^{4} - 87177 T^{5} + 2509711 T^{6} - 87177 p T^{7} + 30443 p^{2} T^{8} - 330 p^{3} T^{9} + 254 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 19 T + 440 T^{2} - 4604 T^{3} + 60189 T^{4} - 431041 T^{5} + 4937144 T^{6} - 431041 p T^{7} + 60189 p^{2} T^{8} - 4604 p^{3} T^{9} + 440 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 15 T + 500 T^{2} + 5412 T^{3} + 100709 T^{4} + 829239 T^{5} + 10974575 T^{6} + 829239 p T^{7} + 100709 p^{2} T^{8} + 5412 p^{3} T^{9} + 500 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 14 T + 431 T^{2} - 3891 T^{3} + 73571 T^{4} - 493163 T^{5} + 7716806 T^{6} - 493163 p T^{7} + 73571 p^{2} T^{8} - 3891 p^{3} T^{9} + 431 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + T + 368 T^{2} - 414 T^{3} + 58751 T^{4} - 162591 T^{5} + 6301840 T^{6} - 162591 p T^{7} + 58751 p^{2} T^{8} - 414 p^{3} T^{9} + 368 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.23894393985650837534461471135, −4.06333021432804633532633271562, −4.02262011158150500625416949266, −3.74313457535651595409557893427, −3.65454087622313029079768751724, −3.49685184631228161865213151070, −3.42485328310591632723444785706, −2.97449936285844218862250774659, −2.90411352145502976649127831712, −2.87758838866718194724186196946, −2.80920937528952476318077630752, −2.74707087198032035896045104076, −2.55914438862766381879261349930, −2.09299988292922068931431924531, −2.08862047376981057293731645325, −1.97033735192791659535094085794, −1.86432434692561275170696201252, −1.61212059337013755370511986562, −1.52576744798469379120805777221, −1.03383820082752558671530135073, −0.793638779721809011715806770661, −0.75960389197591574811017996442, −0.60158734692026153752949617451, −0.58659761012715580114449516196, −0.25395920746196872756643691933, 0.25395920746196872756643691933, 0.58659761012715580114449516196, 0.60158734692026153752949617451, 0.75960389197591574811017996442, 0.793638779721809011715806770661, 1.03383820082752558671530135073, 1.52576744798469379120805777221, 1.61212059337013755370511986562, 1.86432434692561275170696201252, 1.97033735192791659535094085794, 2.08862047376981057293731645325, 2.09299988292922068931431924531, 2.55914438862766381879261349930, 2.74707087198032035896045104076, 2.80920937528952476318077630752, 2.87758838866718194724186196946, 2.90411352145502976649127831712, 2.97449936285844218862250774659, 3.42485328310591632723444785706, 3.49685184631228161865213151070, 3.65454087622313029079768751724, 3.74313457535651595409557893427, 4.02262011158150500625416949266, 4.06333021432804633532633271562, 4.23894393985650837534461471135

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

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