Properties

Label 12-1040e6-1.1-c1e6-0-1
Degree $12$
Conductor $1.265\times 10^{18}$
Sign $1$
Analytic cond. $327991.$
Root an. cond. $2.88174$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·9-s + 12·23-s − 3·25-s − 4·27-s − 12·29-s − 12·43-s + 18·49-s − 12·53-s − 12·61-s + 24·79-s + 9·81-s + 12·101-s − 24·103-s − 60·107-s + 48·113-s + 30·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9·169-s + 173-s + ⋯
L(s)  = 1  − 2·9-s + 2.50·23-s − 3/5·25-s − 0.769·27-s − 2.22·29-s − 1.82·43-s + 18/7·49-s − 1.64·53-s − 1.53·61-s + 2.70·79-s + 81-s + 1.19·101-s − 2.36·103-s − 5.80·107-s + 4.51·113-s + 2.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 9/13·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 5^{6} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(327991.\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1040} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 5^{6} \cdot 13^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.357990554\)
\(L(\frac12)\) \(\approx\) \(1.357990554\)
\(L(\frac{3}{2})\) 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 \( ( 1 + T^{2} )^{3} \)
13 \( 1 - 9 T^{2} - 16 T^{3} - 9 p T^{4} + p^{3} T^{6} \)
good3 \( ( 1 + p T^{2} + 2 T^{3} + p^{2} T^{4} + p^{3} T^{6} )^{2} \)
7 \( 1 - 18 T^{2} + 207 T^{4} - 1676 T^{6} + 207 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 30 T^{2} + 447 T^{4} - 4912 T^{6} + 447 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} \)
17 \( ( 1 + p T^{2} )^{6} \)
19 \( 1 - 18 T^{2} + 423 T^{4} - 200 p T^{6} + 423 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \)
23 \( ( 1 - 6 T + 39 T^{2} - 102 T^{3} + 39 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( ( 1 + 6 T + 75 T^{2} + 264 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 - 126 T^{2} + 7911 T^{4} - 303536 T^{6} + 7911 p^{2} T^{8} - 126 p^{4} T^{10} + p^{6} T^{12} \)
37 \( 1 - 78 T^{2} + 5271 T^{4} - 214292 T^{6} + 5271 p^{2} T^{8} - 78 p^{4} T^{10} + p^{6} T^{12} \)
41 \( 1 - 138 T^{2} + 9663 T^{4} - 461068 T^{6} + 9663 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} \)
43 \( ( 1 + 6 T + 87 T^{2} + 470 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 66 T^{2} + 5055 T^{4} - 172204 T^{6} + 5055 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \)
53 \( ( 1 + 6 T + 75 T^{2} + 660 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 138 T^{2} + 13767 T^{4} - 903112 T^{6} + 13767 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} \)
61 \( ( 1 + 6 T + 87 T^{2} + 200 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 162 T^{2} + 19431 T^{4} - 1405100 T^{6} + 19431 p^{2} T^{8} - 162 p^{4} T^{10} + p^{6} T^{12} \)
71 \( 1 + 6 T^{2} - 417 T^{4} - 224296 T^{6} - 417 p^{2} T^{8} + 6 p^{4} T^{10} + p^{6} T^{12} \)
73 \( 1 - 150 T^{2} + 4479 T^{4} + 230236 T^{6} + 4479 p^{2} T^{8} - 150 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 - 12 T + 93 T^{2} - 200 T^{3} + 93 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 354 T^{2} + 61575 T^{4} - 6424108 T^{6} + 61575 p^{2} T^{8} - 354 p^{4} T^{10} + p^{6} T^{12} \)
89 \( 1 + 42 T^{2} + 10527 T^{4} + 115148 T^{6} + 10527 p^{2} T^{8} + 42 p^{4} T^{10} + p^{6} T^{12} \)
97 \( 1 - 378 T^{2} + 64719 T^{4} - 7267052 T^{6} + 64719 p^{2} T^{8} - 378 p^{4} T^{10} + p^{6} T^{12} \)
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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

−5.27394671859625069334500011943, −5.22012886994153803640302637606, −5.08490838069994134685934006176, −4.70704040593572741343587210945, −4.66066292165129229588555299350, −4.38998616756423445305712676529, −4.38997021838414185833428904590, −4.03964730102890457386113108343, −3.85488805321423240665942762691, −3.66578773686151426631565460671, −3.57999037854292848947951138734, −3.37451461165066725031040200521, −3.19253837355735307458783699750, −3.02915261222446577767822506124, −2.79196807006025048337080165198, −2.64461066942963482326557218119, −2.59751823498744246156671535841, −2.22280652612071376264068446283, −1.93120336970853949022758509559, −1.70070502673456382220951555435, −1.69956839686137881733068151461, −1.31047403035555371636557785297, −0.898801915960010559854677642732, −0.52087717628794454071138264638, −0.25157750600542780009122957912, 0.25157750600542780009122957912, 0.52087717628794454071138264638, 0.898801915960010559854677642732, 1.31047403035555371636557785297, 1.69956839686137881733068151461, 1.70070502673456382220951555435, 1.93120336970853949022758509559, 2.22280652612071376264068446283, 2.59751823498744246156671535841, 2.64461066942963482326557218119, 2.79196807006025048337080165198, 3.02915261222446577767822506124, 3.19253837355735307458783699750, 3.37451461165066725031040200521, 3.57999037854292848947951138734, 3.66578773686151426631565460671, 3.85488805321423240665942762691, 4.03964730102890457386113108343, 4.38997021838414185833428904590, 4.38998616756423445305712676529, 4.66066292165129229588555299350, 4.70704040593572741343587210945, 5.08490838069994134685934006176, 5.22012886994153803640302637606, 5.27394671859625069334500011943

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

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