Properties

Label 12-475e6-1.1-c1e6-0-1
Degree $12$
Conductor $1.149\times 10^{16}$
Sign $1$
Analytic cond. $2977.31$
Root an. cond. $1.94753$
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  − 2·4-s − 2·9-s + 2·11-s − 3·16-s − 6·19-s + 36·29-s + 4·36-s + 12·41-s − 4·44-s − 23·49-s + 20·59-s − 14·61-s + 8·64-s + 52·71-s + 12·76-s − 24·79-s + 81-s + 24·89-s − 4·99-s + 20·101-s + 40·109-s − 72·116-s − 31·121-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 4-s − 2/3·9-s + 0.603·11-s − 3/4·16-s − 1.37·19-s + 6.68·29-s + 2/3·36-s + 1.87·41-s − 0.603·44-s − 3.28·49-s + 2.60·59-s − 1.79·61-s + 64-s + 6.17·71-s + 1.37·76-s − 2.70·79-s + 1/9·81-s + 2.54·89-s − 0.402·99-s + 1.99·101-s + 3.83·109-s − 6.68·116-s − 2.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(5^{12} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(2977.31\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{475} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 5^{12} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.511054389\)
\(L(\frac12)\) \(\approx\) \(2.511054389\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( ( 1 + T )^{6} \)
good2 \( 1 + p T^{2} + 7 T^{4} + 3 p^{2} T^{6} + 7 p^{2} T^{8} + p^{5} T^{10} + p^{6} T^{12} \)
3 \( 1 + 2 T^{2} + p T^{4} + 20 T^{6} + p^{3} T^{8} + 2 p^{4} T^{10} + p^{6} T^{12} \)
7 \( 1 + 23 T^{2} + 307 T^{4} + 2586 T^{6} + 307 p^{2} T^{8} + 23 p^{4} T^{10} + p^{6} T^{12} \)
11 \( ( 1 - T + 17 T^{2} - 10 T^{3} + 17 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 + 50 T^{2} + 1315 T^{4} + 21108 T^{6} + 1315 p^{2} T^{8} + 50 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 + 43 T^{2} + 1331 T^{4} + 25042 T^{6} + 1331 p^{2} T^{8} + 43 p^{4} T^{10} + p^{6} T^{12} \)
23 \( 1 + 102 T^{2} + 4831 T^{4} + 138580 T^{6} + 4831 p^{2} T^{8} + 102 p^{4} T^{10} + p^{6} T^{12} \)
29 \( ( 1 - 6 T + p T^{2} )^{6} \)
31 \( ( 1 + 37 T^{2} + 128 T^{3} + 37 p T^{4} + p^{3} T^{6} )^{2} \)
37 \( 1 + 166 T^{2} + 13011 T^{4} + 608316 T^{6} + 13011 p^{2} T^{8} + 166 p^{4} T^{10} + p^{6} T^{12} \)
41 \( ( 1 - 6 T + 79 T^{2} - 516 T^{3} + 79 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( 1 + 239 T^{2} + 24571 T^{4} + 1388154 T^{6} + 24571 p^{2} T^{8} + 239 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 + 95 T^{2} + 5443 T^{4} + 214314 T^{6} + 5443 p^{2} T^{8} + 95 p^{4} T^{10} + p^{6} T^{12} \)
53 \( 1 + 162 T^{2} + 11539 T^{4} + 610708 T^{6} + 11539 p^{2} T^{8} + 162 p^{4} T^{10} + p^{6} T^{12} \)
59 \( ( 1 - 10 T + 185 T^{2} - 1132 T^{3} + 185 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 + 7 T + 79 T^{2} + 78 T^{3} + 79 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 + 62 T^{2} + 4771 T^{4} + 199788 T^{6} + 4771 p^{2} T^{8} + 62 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 - 26 T + 413 T^{2} - 4124 T^{3} + 413 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 + 307 T^{2} + 43299 T^{4} + 3822498 T^{6} + 43299 p^{2} T^{8} + 307 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 + 12 T + 229 T^{2} + 1864 T^{3} + 229 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 + 270 T^{2} + 39367 T^{4} + 3817060 T^{6} + 39367 p^{2} T^{8} + 270 p^{4} T^{10} + p^{6} T^{12} \)
89 \( ( 1 - 12 T - 17 T^{2} + 1320 T^{3} - 17 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 + 554 T^{2} + 130507 T^{4} + 16717956 T^{6} + 130507 p^{2} T^{8} + 554 p^{4} T^{10} + p^{6} T^{12} \)
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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

−6.04040376453593740695364213529, −5.94699259565334212563688337217, −5.41661366155861086766703204900, −5.39421644333512201886752838619, −5.16646578318485492417535539239, −4.91908694619561831623350060779, −4.81723383056450431016870446997, −4.61746627410580203861480523527, −4.55690299393144658623841334770, −4.44968527760017289098763587493, −4.21181576347591212337222857738, −4.00445050697294118143489854704, −3.74002195520786417313797068315, −3.45307914098692482568328632010, −3.23585636369928182955420610606, −3.20032577494930613732152515152, −2.73889190584271940849123491468, −2.62838815475436065308179961045, −2.49386011183497425266052928076, −2.15062333303143639152485539722, −1.93948977006853465589612621317, −1.57096617156208095214207025588, −0.937803168317371011469960140801, −0.804271158418313148776300199802, −0.55456626669807145697844161379, 0.55456626669807145697844161379, 0.804271158418313148776300199802, 0.937803168317371011469960140801, 1.57096617156208095214207025588, 1.93948977006853465589612621317, 2.15062333303143639152485539722, 2.49386011183497425266052928076, 2.62838815475436065308179961045, 2.73889190584271940849123491468, 3.20032577494930613732152515152, 3.23585636369928182955420610606, 3.45307914098692482568328632010, 3.74002195520786417313797068315, 4.00445050697294118143489854704, 4.21181576347591212337222857738, 4.44968527760017289098763587493, 4.55690299393144658623841334770, 4.61746627410580203861480523527, 4.81723383056450431016870446997, 4.91908694619561831623350060779, 5.16646578318485492417535539239, 5.39421644333512201886752838619, 5.41661366155861086766703204900, 5.94699259565334212563688337217, 6.04040376453593740695364213529

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

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