Properties

Label 12-5225e6-1.1-c1e6-0-0
Degree $12$
Conductor $2.035\times 10^{22}$
Sign $1$
Analytic cond. $5.27448\times 10^{9}$
Root an. cond. $6.45924$
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  − 3·3-s − 2·4-s − 5·7-s − 6·11-s + 6·12-s + 9·13-s − 16-s + 5·17-s − 6·19-s + 15·21-s − 8·23-s + 2·27-s + 10·28-s − 5·29-s − 31-s − 3·32-s + 18·33-s − 9·37-s − 27·39-s + 25·41-s − 15·43-s + 12·44-s − 24·47-s + 3·48-s − 2·49-s − 15·51-s − 18·52-s + ⋯
L(s)  = 1  − 1.73·3-s − 4-s − 1.88·7-s − 1.80·11-s + 1.73·12-s + 2.49·13-s − 1/4·16-s + 1.21·17-s − 1.37·19-s + 3.27·21-s − 1.66·23-s + 0.384·27-s + 1.88·28-s − 0.928·29-s − 0.179·31-s − 0.530·32-s + 3.13·33-s − 1.47·37-s − 4.32·39-s + 3.90·41-s − 2.28·43-s + 1.80·44-s − 3.50·47-s + 0.433·48-s − 2/7·49-s − 2.10·51-s − 2.49·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 11^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 11^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(5^{12} \cdot 11^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(5.27448\times 10^{9}\)
Root analytic conductor: \(6.45924\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 5^{12} \cdot 11^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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)$
bad5 \( 1 \)
11 \( ( 1 + T )^{6} \)
19 \( ( 1 + T )^{6} \)
good2 \( 1 + p T^{2} + 5 T^{4} + 3 T^{5} + 11 T^{6} + 3 p T^{7} + 5 p^{2} T^{8} + p^{5} T^{10} + p^{6} T^{12} \)
3 \( 1 + p T + p^{2} T^{2} + 25 T^{3} + 2 p^{3} T^{4} + 35 p T^{5} + 200 T^{6} + 35 p^{2} T^{7} + 2 p^{5} T^{8} + 25 p^{3} T^{9} + p^{6} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
7 \( 1 + 5 T + 27 T^{2} + 85 T^{3} + 260 T^{4} + 641 T^{5} + 1664 T^{6} + 641 p T^{7} + 260 p^{2} T^{8} + 85 p^{3} T^{9} + 27 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 9 T + 87 T^{2} - 522 T^{3} + 2961 T^{4} - 12816 T^{5} + 51960 T^{6} - 12816 p T^{7} + 2961 p^{2} T^{8} - 522 p^{3} T^{9} + 87 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 5 T + 3 p T^{2} - 236 T^{3} + 1803 T^{4} - 6582 T^{5} + 2080 p T^{6} - 6582 p T^{7} + 1803 p^{2} T^{8} - 236 p^{3} T^{9} + 3 p^{5} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 8 T + 122 T^{2} + 797 T^{3} + 6515 T^{4} + 34306 T^{5} + 8480 p T^{6} + 34306 p T^{7} + 6515 p^{2} T^{8} + 797 p^{3} T^{9} + 122 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 5 T + 113 T^{2} + 278 T^{3} + 4691 T^{4} + 2962 T^{5} + 131272 T^{6} + 2962 p T^{7} + 4691 p^{2} T^{8} + 278 p^{3} T^{9} + 113 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + T + 125 T^{2} + 110 T^{3} + 7805 T^{4} + 6154 T^{5} + 300002 T^{6} + 6154 p T^{7} + 7805 p^{2} T^{8} + 110 p^{3} T^{9} + 125 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 9 T + 182 T^{2} + 1159 T^{3} + 13932 T^{4} + 69013 T^{5} + 635864 T^{6} + 69013 p T^{7} + 13932 p^{2} T^{8} + 1159 p^{3} T^{9} + 182 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 25 T + 366 T^{2} - 4144 T^{3} + 39720 T^{4} - 316980 T^{5} + 2173916 T^{6} - 316980 p T^{7} + 39720 p^{2} T^{8} - 4144 p^{3} T^{9} + 366 p^{4} T^{10} - 25 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 15 T + 295 T^{2} + 2949 T^{3} + 33100 T^{4} + 242797 T^{5} + 1919164 T^{6} + 242797 p T^{7} + 33100 p^{2} T^{8} + 2949 p^{3} T^{9} + 295 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 24 T + 378 T^{2} + 4753 T^{3} + 48513 T^{4} + 418146 T^{5} + 3090596 T^{6} + 418146 p T^{7} + 48513 p^{2} T^{8} + 4753 p^{3} T^{9} + 378 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 5 T - 3 T^{2} - 306 T^{3} + 2769 T^{4} - 2336 T^{5} - 19036 T^{6} - 2336 p T^{7} + 2769 p^{2} T^{8} - 306 p^{3} T^{9} - 3 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 39 T + 967 T^{2} - 16440 T^{3} + 218093 T^{4} - 2276228 T^{5} + 19448898 T^{6} - 2276228 p T^{7} + 218093 p^{2} T^{8} - 16440 p^{3} T^{9} + 967 p^{4} T^{10} - 39 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 11 T + 213 T^{2} + 2208 T^{3} + 27847 T^{4} + 221366 T^{5} + 2110000 T^{6} + 221366 p T^{7} + 27847 p^{2} T^{8} + 2208 p^{3} T^{9} + 213 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 24 T + 415 T^{2} + 4488 T^{3} + 38107 T^{4} + 254713 T^{5} + 1858810 T^{6} + 254713 p T^{7} + 38107 p^{2} T^{8} + 4488 p^{3} T^{9} + 415 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 24 T + 378 T^{2} + 2955 T^{3} + 10575 T^{4} - 126768 T^{5} - 1650364 T^{6} - 126768 p T^{7} + 10575 p^{2} T^{8} + 2955 p^{3} T^{9} + 378 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 26 T + 441 T^{2} - 5682 T^{3} + 64207 T^{4} - 616835 T^{5} + 5486980 T^{6} - 616835 p T^{7} + 64207 p^{2} T^{8} - 5682 p^{3} T^{9} + 441 p^{4} T^{10} - 26 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 11 T + 456 T^{2} - 3763 T^{3} + 86160 T^{4} - 549915 T^{5} + 8915718 T^{6} - 549915 p T^{7} + 86160 p^{2} T^{8} - 3763 p^{3} T^{9} + 456 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 39 T + 763 T^{2} + 10182 T^{3} + 114167 T^{4} + 1204688 T^{5} + 11682726 T^{6} + 1204688 p T^{7} + 114167 p^{2} T^{8} + 10182 p^{3} T^{9} + 763 p^{4} T^{10} + 39 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 22 T + 555 T^{2} - 8566 T^{3} + 122441 T^{4} - 1418929 T^{5} + 14412680 T^{6} - 1418929 p T^{7} + 122441 p^{2} T^{8} - 8566 p^{3} T^{9} + 555 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 22 T + 449 T^{2} + 6218 T^{3} + 77889 T^{4} + 800297 T^{5} + 8487036 T^{6} + 800297 p T^{7} + 77889 p^{2} T^{8} + 6218 p^{3} T^{9} + 449 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.72937850190224331406413743471, −4.56265659441558939684051982689, −4.15869849638887161412617317454, −4.15189801884159778289470515593, −4.05099374802824709313475731003, −3.87556241984402427231238466473, −3.85884512913562412324710752762, −3.63781285720558863062882860005, −3.50751269787072095913438281317, −3.47796297777766996595124710141, −3.34234165260477157637872816565, −3.09205466247752282732067794925, −2.95458918729745372071998955628, −2.72171306558403445201561688595, −2.61881748081244937500683330322, −2.57947618940417147967218347496, −2.41824787307215002851607943679, −2.09331058939856502589404223913, −1.99078771617525917834672030086, −1.68360697737682742178596023455, −1.49417526420053983440673895278, −1.48857561925687474626953134017, −1.23436648291945867257609904186, −0.916759691176279815755377172468, −0.908605004971853356444654691527, 0, 0, 0, 0, 0, 0, 0.908605004971853356444654691527, 0.916759691176279815755377172468, 1.23436648291945867257609904186, 1.48857561925687474626953134017, 1.49417526420053983440673895278, 1.68360697737682742178596023455, 1.99078771617525917834672030086, 2.09331058939856502589404223913, 2.41824787307215002851607943679, 2.57947618940417147967218347496, 2.61881748081244937500683330322, 2.72171306558403445201561688595, 2.95458918729745372071998955628, 3.09205466247752282732067794925, 3.34234165260477157637872816565, 3.47796297777766996595124710141, 3.50751269787072095913438281317, 3.63781285720558863062882860005, 3.85884512913562412324710752762, 3.87556241984402427231238466473, 4.05099374802824709313475731003, 4.15189801884159778289470515593, 4.15869849638887161412617317454, 4.56265659441558939684051982689, 4.72937850190224331406413743471

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

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