Properties

Label 12-5070e6-1.1-c1e6-0-2
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

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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 + 6·17-s + 18·23-s − 3·25-s + 56·27-s − 63·36-s + 6·43-s + 36·48-s + 4·49-s + 36·51-s − 34·53-s + 22·61-s − 10·64-s − 18·68-s + 108·69-s − 18·75-s − 6·79-s + 126·81-s − 54·92-s + 9·100-s + 28·101-s + 2·103-s + 46·107-s + ⋯
L(s)  = 1  + 3.46·3-s − 3/2·4-s + 7·9-s − 5.19·12-s + 3/2·16-s + 1.45·17-s + 3.75·23-s − 3/5·25-s + 10.7·27-s − 10.5·36-s + 0.914·43-s + 5.19·48-s + 4/7·49-s + 5.04·51-s − 4.67·53-s + 2.81·61-s − 5/4·64-s − 2.18·68-s + 13.0·69-s − 2.07·75-s − 0.675·79-s + 14·81-s − 5.62·92-s + 9/10·100-s + 2.78·101-s + 0.197·103-s + 4.44·107-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

\(L(1)\) \(\approx\) \(10.12443605\)
\(L(\frac12)\) \(\approx\) \(10.12443605\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T^{2} + 52 T^{4} - 181 T^{6} + 52 p^{2} T^{8} - 4 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 40 T^{2} + 824 T^{4} - 10941 T^{6} + 824 p^{2} T^{8} - 40 p^{4} T^{10} + p^{6} T^{12} \)
17 \( ( 1 - 3 T + 5 T^{2} + 37 T^{3} + 5 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( 1 + 15 T^{2} + 269 T^{4} + 9751 T^{6} + 269 p^{2} T^{8} + 15 p^{4} T^{10} + p^{6} T^{12} \)
23 \( ( 1 - 9 T + 89 T^{2} - 427 T^{3} + 89 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( ( 1 + 24 T^{2} + 189 T^{3} + 24 p T^{4} + p^{3} T^{6} )^{2} \)
31 \( 1 - 96 T^{2} + 5696 T^{4} - 208901 T^{6} + 5696 p^{2} T^{8} - 96 p^{4} T^{10} + p^{6} T^{12} \)
37 \( 1 - 100 T^{2} + 7060 T^{4} - 300217 T^{6} + 7060 p^{2} T^{8} - 100 p^{4} T^{10} + p^{6} T^{12} \)
41 \( 1 - 149 T^{2} + 12273 T^{4} - 615401 T^{6} + 12273 p^{2} T^{8} - 149 p^{4} T^{10} + p^{6} T^{12} \)
43 \( ( 1 - 3 T + 69 T^{2} - 385 T^{3} + 69 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 55 T^{2} + 5694 T^{4} - 187411 T^{6} + 5694 p^{2} T^{8} - 55 p^{4} T^{10} + p^{6} T^{12} \)
53 \( ( 1 + 17 T + 239 T^{2} + 1873 T^{3} + 239 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - T^{2} + 6101 T^{4} + 40527 T^{6} + 6101 p^{2} T^{8} - p^{4} T^{10} + p^{6} T^{12} \)
61 \( ( 1 - 11 T + 193 T^{2} - 1313 T^{3} + 193 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 3 p T^{2} + 13641 T^{4} - 624841 T^{6} + 13641 p^{2} T^{8} - 3 p^{5} T^{10} + p^{6} T^{12} \)
71 \( 1 - 197 T^{2} + 11817 T^{4} - 407513 T^{6} + 11817 p^{2} T^{8} - 197 p^{4} T^{10} + p^{6} T^{12} \)
73 \( 1 + 43 T^{2} - 3671 T^{4} - 558233 T^{6} - 3671 p^{2} T^{8} + 43 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 + 3 T + 219 T^{2} + 447 T^{3} + 219 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 408 T^{2} + 74832 T^{4} - 7943533 T^{6} + 74832 p^{2} T^{8} - 408 p^{4} T^{10} + p^{6} T^{12} \)
89 \( 1 - 53 T^{2} + 7197 T^{4} - 833297 T^{6} + 7197 p^{2} T^{8} - 53 p^{4} T^{10} + p^{6} T^{12} \)
97 \( 1 + 32 T^{2} + 17156 T^{4} + 802935 T^{6} + 17156 p^{2} T^{8} + 32 p^{4} T^{10} + p^{6} T^{12} \)
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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

−4.26458949944005174961018015484, −3.88680264408018960750061816419, −3.83363572945920987749622210381, −3.79966808323726158849833781461, −3.46659737160920582675852300934, −3.44032922867463791936741232613, −3.37317424653556137921514007711, −3.29381609588877298272604138672, −3.12969651618580874200166212718, −3.01903863790114294665378567774, −2.96564075642220655788875884893, −2.56835891833709634066077370664, −2.44882212093264362702222610065, −2.44782463179688390390343418303, −2.25270437277617276257157482121, −2.09394379201956295066445672341, −1.96173845377314467885795007339, −1.52404693768390337656619097040, −1.51037591269071385119672052316, −1.24307363973396128094039139024, −1.17181104551611772204828101335, −1.07481664061439415025040204479, −0.77554403893384420728337870847, −0.62061807626077117394380634180, −0.13368404916328274429599631875, 0.13368404916328274429599631875, 0.62061807626077117394380634180, 0.77554403893384420728337870847, 1.07481664061439415025040204479, 1.17181104551611772204828101335, 1.24307363973396128094039139024, 1.51037591269071385119672052316, 1.52404693768390337656619097040, 1.96173845377314467885795007339, 2.09394379201956295066445672341, 2.25270437277617276257157482121, 2.44782463179688390390343418303, 2.44882212093264362702222610065, 2.56835891833709634066077370664, 2.96564075642220655788875884893, 3.01903863790114294665378567774, 3.12969651618580874200166212718, 3.29381609588877298272604138672, 3.37317424653556137921514007711, 3.44032922867463791936741232613, 3.46659737160920582675852300934, 3.79966808323726158849833781461, 3.83363572945920987749622210381, 3.88680264408018960750061816419, 4.26458949944005174961018015484

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

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