Properties

Label 12-5070e6-1.1-c1e6-0-0
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 − 6·17-s − 22·23-s − 3·25-s − 56·27-s + 4·29-s − 63·36-s + 10·43-s − 36·48-s + 16·49-s + 36·51-s + 38·53-s − 18·61-s − 10·64-s + 18·68-s + 132·69-s + 18·75-s − 50·79-s + 126·81-s − 24·87-s + 66·92-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 − 1.45·17-s − 4.58·23-s − 3/5·25-s − 10.7·27-s + 0.742·29-s − 10.5·36-s + 1.52·43-s − 5.19·48-s + 16/7·49-s + 5.04·51-s + 5.21·53-s − 2.30·61-s − 5/4·64-s + 2.18·68-s + 15.8·69-s + 2.07·75-s − 5.62·79-s + 14·81-s − 2.57·87-s + 6.88·92-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

\(L(1)\) \(\approx\) \(0.003201817922\)
\(L(\frac12)\) \(\approx\) \(0.003201817922\)
\(L(\frac{3}{2})\) 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 \( 1 - 16 T^{2} + 160 T^{4} - 1189 T^{6} + 160 p^{2} T^{8} - 16 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 60 T^{2} + 1556 T^{4} - 22373 T^{6} + 1556 p^{2} T^{8} - 60 p^{4} T^{10} + p^{6} T^{12} \)
17 \( ( 1 + 3 T + 5 T^{2} + 5 T^{3} + 5 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( 1 - 69 T^{2} + 2481 T^{4} - 57449 T^{6} + 2481 p^{2} T^{8} - 69 p^{4} T^{10} + p^{6} T^{12} \)
23 \( ( 1 + 11 T + 79 T^{2} + 393 T^{3} + 79 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( ( 1 - 2 T + 44 T^{2} + 11 T^{3} + 44 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 - 172 T^{2} + 12728 T^{4} - 518085 T^{6} + 12728 p^{2} T^{8} - 172 p^{4} T^{10} + p^{6} T^{12} \)
37 \( 1 + 28 T^{2} + 4072 T^{4} + 72751 T^{6} + 4072 p^{2} T^{8} + 28 p^{4} T^{10} + p^{6} T^{12} \)
41 \( 1 - 157 T^{2} + 11465 T^{4} - 549297 T^{6} + 11465 p^{2} T^{8} - 157 p^{4} T^{10} + p^{6} T^{12} \)
43 \( ( 1 - 5 T + 121 T^{2} - 431 T^{3} + 121 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 + 89 T^{2} + 5870 T^{4} + 246861 T^{6} + 5870 p^{2} T^{8} + 89 p^{4} T^{10} + p^{6} T^{12} \)
53 \( ( 1 - 19 T + 207 T^{2} - 1665 T^{3} + 207 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 117 T^{2} + 9917 T^{4} - 738521 T^{6} + 9917 p^{2} T^{8} - 117 p^{4} T^{10} + p^{6} T^{12} \)
61 \( ( 1 + 9 T + 161 T^{2} + 1069 T^{3} + 161 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 257 T^{2} + 34417 T^{4} - 2856329 T^{6} + 34417 p^{2} T^{8} - 257 p^{4} T^{10} + p^{6} T^{12} \)
71 \( 1 - 409 T^{2} + 70881 T^{4} - 6657217 T^{6} + 70881 p^{2} T^{8} - 409 p^{4} T^{10} + p^{6} T^{12} \)
73 \( 1 - 253 T^{2} + 27157 T^{4} - 2044633 T^{6} + 27157 p^{2} T^{8} - 253 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 + 25 T + 387 T^{2} + 4075 T^{3} + 387 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 136 T^{2} - 1408 T^{4} + 1106091 T^{6} - 1408 p^{2} T^{8} - 136 p^{4} T^{10} + p^{6} T^{12} \)
89 \( 1 - 37 T^{2} + 16461 T^{4} - 666793 T^{6} + 16461 p^{2} T^{8} - 37 p^{4} T^{10} + p^{6} T^{12} \)
97 \( 1 - 444 T^{2} + 93876 T^{4} - 11587849 T^{6} + 93876 p^{2} T^{8} - 444 p^{4} T^{10} + p^{6} T^{12} \)
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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

−4.27524193047029336613015669705, −4.13603556917027478751956895824, −4.02378166887931148281603280304, −3.93517234554547645527766715621, −3.75673862912329425052465037664, −3.72299465215407689294851324269, −3.71918091038490932846251341867, −3.39348862016188157831533201412, −3.13823860564336293024844275985, −2.86104450533343174208273995289, −2.74051778543650921258151942529, −2.47520065438442194037995021589, −2.35267125472025301286286419469, −2.23190292054828369381780175046, −2.16725424389226685184917412721, −2.05632927727596093598042402563, −1.63884735193195966753317579827, −1.47992159284891512570411374119, −1.31151364500920958810504746257, −1.16219775721449570606014574495, −1.02270348373276534221887753860, −0.66396435811652807087444354247, −0.55403118892870677921553223236, −0.20329314988200176488010316024, −0.02631371379964150618741422906, 0.02631371379964150618741422906, 0.20329314988200176488010316024, 0.55403118892870677921553223236, 0.66396435811652807087444354247, 1.02270348373276534221887753860, 1.16219775721449570606014574495, 1.31151364500920958810504746257, 1.47992159284891512570411374119, 1.63884735193195966753317579827, 2.05632927727596093598042402563, 2.16725424389226685184917412721, 2.23190292054828369381780175046, 2.35267125472025301286286419469, 2.47520065438442194037995021589, 2.74051778543650921258151942529, 2.86104450533343174208273995289, 3.13823860564336293024844275985, 3.39348862016188157831533201412, 3.71918091038490932846251341867, 3.72299465215407689294851324269, 3.75673862912329425052465037664, 3.93517234554547645527766715621, 4.02378166887931148281603280304, 4.13603556917027478751956895824, 4.27524193047029336613015669705

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

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