Properties

Label 12-5070e6-1.1-c1e6-0-4
Degree $12$
Conductor $1.698\times 10^{22}$
Sign $1$
Analytic cond. $4.40261\times 10^{9}$
Root an. cond. $6.36271$
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  + 6·3-s − 3·4-s + 21·9-s − 18·12-s + 6·16-s + 14·17-s − 6·23-s − 3·25-s + 56·27-s − 28·29-s − 63·36-s − 18·43-s + 36·48-s + 16·49-s + 84·51-s + 6·53-s − 42·61-s − 10·64-s − 42·68-s − 36·69-s − 18·75-s + 30·79-s + 126·81-s − 168·87-s + 18·92-s + 9·100-s − 12·101-s + ⋯
L(s)  = 1  + 3.46·3-s − 3/2·4-s + 7·9-s − 5.19·12-s + 3/2·16-s + 3.39·17-s − 1.25·23-s − 3/5·25-s + 10.7·27-s − 5.19·29-s − 10.5·36-s − 2.74·43-s + 5.19·48-s + 16/7·49-s + 11.7·51-s + 0.824·53-s − 5.37·61-s − 5/4·64-s − 5.09·68-s − 4.33·69-s − 2.07·75-s + 3.37·79-s + 14·81-s − 18.0·87-s + 1.87·92-s + 9/10·100-s − 1.19·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(4.40261\times 10^{9}\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5070} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(13.52983307\)
\(L(\frac12)\) \(\approx\) \(13.52983307\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{3} \)
3 \( ( 1 - T )^{6} \)
5 \( ( 1 + T^{2} )^{3} \)
13 \( 1 \)
good7 \( 1 - 16 T^{2} + 20 T^{4} + 603 T^{6} + 20 p^{2} T^{8} - 16 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 24 T^{2} + 408 T^{4} - 5781 T^{6} + 408 p^{2} T^{8} - 24 p^{4} T^{10} + p^{6} T^{12} \)
17 \( ( 1 - 7 T + 3 p T^{2} - 231 T^{3} + 3 p^{2} T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( 1 - 53 T^{2} + 1849 T^{4} - 40673 T^{6} + 1849 p^{2} T^{8} - 53 p^{4} T^{10} + p^{6} T^{12} \)
23 \( ( 1 + 3 T + 65 T^{2} + 125 T^{3} + 65 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( ( 1 + 14 T + 136 T^{2} + 819 T^{3} + 136 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 - 120 T^{2} + 6500 T^{4} - 231941 T^{6} + 6500 p^{2} T^{8} - 120 p^{4} T^{10} + p^{6} T^{12} \)
37 \( 1 - 84 T^{2} + 6396 T^{4} - 250369 T^{6} + 6396 p^{2} T^{8} - 84 p^{4} T^{10} + p^{6} T^{12} \)
41 \( 1 - 201 T^{2} + 18237 T^{4} - 957345 T^{6} + 18237 p^{2} T^{8} - 201 p^{4} T^{10} + p^{6} T^{12} \)
43 \( ( 1 + 9 T + 149 T^{2} + 773 T^{3} + 149 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( ( 1 - 45 T^{2} + p^{2} T^{4} )^{3} \)
53 \( ( 1 - 3 T + 29 T^{2} - 95 T^{3} + 29 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 + 31 T^{2} + 7317 T^{4} + 115303 T^{6} + 7317 p^{2} T^{8} + 31 p^{4} T^{10} + p^{6} T^{12} \)
61 \( ( 1 + 21 T + 267 T^{2} + 2275 T^{3} + 267 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 165 T^{2} + 17453 T^{4} - 1430969 T^{6} + 17453 p^{2} T^{8} - 165 p^{4} T^{10} + p^{6} T^{12} \)
71 \( 1 - 265 T^{2} + 37241 T^{4} - 3241713 T^{6} + 37241 p^{2} T^{8} - 265 p^{4} T^{10} + p^{6} T^{12} \)
73 \( 1 - 145 T^{2} + 10673 T^{4} - 711921 T^{6} + 10673 p^{2} T^{8} - 145 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 - 15 T + 249 T^{2} - 2369 T^{3} + 249 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 4 p T^{2} + 53472 T^{4} - 5401109 T^{6} + 53472 p^{2} T^{8} - 4 p^{5} T^{10} + p^{6} T^{12} \)
89 \( 1 - 237 T^{2} + 30873 T^{4} - 2909697 T^{6} + 30873 p^{2} T^{8} - 237 p^{4} T^{10} + p^{6} T^{12} \)
97 \( 1 - 296 T^{2} + 45208 T^{4} - 4811201 T^{6} + 45208 p^{2} T^{8} - 296 p^{4} T^{10} + p^{6} T^{12} \)
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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

−3.98902694939015341084779198709, −3.80762054526566905777518459465, −3.79727425631430690649028525444, −3.77229790293537996765107911806, −3.72960354445876380533247773354, −3.48084904034790716860825364281, −3.44973874229984704262759082524, −3.25507157292930185346060352037, −3.20418376365781325115226018298, −2.99882006160799491865799085244, −2.84500305633886341567744098985, −2.73594018512066361243243007140, −2.58572187856610350622794271558, −2.28313307707080526488448093929, −2.01326901708410691471859695820, −1.93807948021323121121811677066, −1.84838658611531074187978966018, −1.79362052713613289294613167057, −1.66638891245132606962745416442, −1.37162365133848167514726762271, −1.21260996413552044409472272698, −1.02989473914475097512204666517, −0.65979283773728119263811915740, −0.45164107774321531685043783503, −0.21635385889549210970610749489, 0.21635385889549210970610749489, 0.45164107774321531685043783503, 0.65979283773728119263811915740, 1.02989473914475097512204666517, 1.21260996413552044409472272698, 1.37162365133848167514726762271, 1.66638891245132606962745416442, 1.79362052713613289294613167057, 1.84838658611531074187978966018, 1.93807948021323121121811677066, 2.01326901708410691471859695820, 2.28313307707080526488448093929, 2.58572187856610350622794271558, 2.73594018512066361243243007140, 2.84500305633886341567744098985, 2.99882006160799491865799085244, 3.20418376365781325115226018298, 3.25507157292930185346060352037, 3.44973874229984704262759082524, 3.48084904034790716860825364281, 3.72960354445876380533247773354, 3.77229790293537996765107911806, 3.79727425631430690649028525444, 3.80762054526566905777518459465, 3.98902694939015341084779198709

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

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