Properties

Label 12-1280e6-1.1-c1e6-0-3
Degree $12$
Conductor $4.398\times 10^{18}$
Sign $1$
Analytic cond. $1.14004\times 10^{6}$
Root an. cond. $3.19700$
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  + 4·3-s + 2·9-s + 16·13-s + 25-s − 12·27-s + 16·31-s + 16·37-s + 64·39-s − 4·41-s − 28·43-s + 18·49-s − 32·53-s − 20·67-s + 24·71-s + 4·75-s − 32·79-s − 19·81-s + 60·83-s − 20·89-s + 64·93-s − 20·107-s + 64·111-s + 32·117-s + 22·121-s − 16·123-s − 16·125-s + 127-s + ⋯
L(s)  = 1  + 2.30·3-s + 2/3·9-s + 4.43·13-s + 1/5·25-s − 2.30·27-s + 2.87·31-s + 2.63·37-s + 10.2·39-s − 0.624·41-s − 4.26·43-s + 18/7·49-s − 4.39·53-s − 2.44·67-s + 2.84·71-s + 0.461·75-s − 3.60·79-s − 2.11·81-s + 6.58·83-s − 2.11·89-s + 6.63·93-s − 1.93·107-s + 6.07·111-s + 2.95·117-s + 2·121-s − 1.44·123-s − 1.43·125-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{48} \cdot 5^{6}\)
Sign: $1$
Analytic conductor: \(1.14004\times 10^{6}\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{48} \cdot 5^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(19.84520101\)
\(L(\frac12)\) \(\approx\) \(19.84520101\)
\(L(\frac{3}{2})\) 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 \( 1 - T^{2} + 16 T^{3} - p T^{4} + p^{3} T^{6} \)
good3 \( ( 1 - 2 T + 5 T^{2} - 8 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
7 \( 1 - 18 T^{2} + 239 T^{4} - 1868 T^{6} + 239 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 2 p T^{2} + 311 T^{4} - 4116 T^{6} + 311 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \)
13 \( ( 1 - 8 T + 47 T^{2} - 192 T^{3} + 47 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 - 54 T^{2} + 1583 T^{4} - 31412 T^{6} + 1583 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} \)
19 \( 1 - 2 p T^{2} + 1351 T^{4} - 26804 T^{6} + 1351 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \)
23 \( 1 - 98 T^{2} + 4335 T^{4} - 120044 T^{6} + 4335 p^{2} T^{8} - 98 p^{4} T^{10} + p^{6} T^{12} \)
29 \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{3} \)
31 \( ( 1 - 8 T + 61 T^{2} - 368 T^{3} + 61 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( ( 1 - 8 T + 79 T^{2} - 320 T^{3} + 79 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( ( 1 + 2 T + 71 T^{2} - 20 T^{3} + 71 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 + 14 T + 149 T^{2} + 1104 T^{3} + 149 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 210 T^{2} + 21311 T^{4} - 1269644 T^{6} + 21311 p^{2} T^{8} - 210 p^{4} T^{10} + p^{6} T^{12} \)
53 \( ( 1 + 16 T + 231 T^{2} + 1776 T^{3} + 231 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 278 T^{2} + 35991 T^{4} - 2711444 T^{6} + 35991 p^{2} T^{8} - 278 p^{4} T^{10} + p^{6} T^{12} \)
61 \( 1 - 210 T^{2} + 18695 T^{4} - 1171868 T^{6} + 18695 p^{2} T^{8} - 210 p^{4} T^{10} + p^{6} T^{12} \)
67 \( ( 1 + 10 T + 141 T^{2} + 736 T^{3} + 141 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
71 \( ( 1 - 12 T + 197 T^{2} - 1384 T^{3} + 197 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 38 T^{2} + 10495 T^{4} - 503444 T^{6} + 10495 p^{2} T^{8} - 38 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 + 16 T + 269 T^{2} + 2400 T^{3} + 269 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( ( 1 - 30 T + 509 T^{2} - 5504 T^{3} + 509 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( ( 1 + 10 T + 151 T^{2} + 684 T^{3} + 151 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 278 T^{2} + 46735 T^{4} - 5290868 T^{6} + 46735 p^{2} T^{8} - 278 p^{4} T^{10} + p^{6} T^{12} \)
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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

−5.05229496815414267386762718627, −4.95663466285069125706354709223, −4.77423615118698125310883709925, −4.45228163340157235125809958289, −4.23425116868559581170039884697, −4.22482062425475065954444429449, −4.17711065949868828021311528243, −3.99378935399928321483722585588, −3.46454143679001584635421554695, −3.43395750938636894574526573694, −3.41603409681711236840952923078, −3.37839948092828449855903856871, −3.10764907866828550411708338943, −2.86055837631765748405711725966, −2.81052550495562783546795003995, −2.75738206584034249570609290386, −2.47689149790938805736249372421, −2.17301472151615180443400491619, −1.77303901588233155962244833102, −1.69690758459532436636789097752, −1.62022215099277919971756635414, −1.35341684850600770582161312430, −0.958793902544414575486056163560, −0.69442204614492528242168396597, −0.50006639441712017858936159212, 0.50006639441712017858936159212, 0.69442204614492528242168396597, 0.958793902544414575486056163560, 1.35341684850600770582161312430, 1.62022215099277919971756635414, 1.69690758459532436636789097752, 1.77303901588233155962244833102, 2.17301472151615180443400491619, 2.47689149790938805736249372421, 2.75738206584034249570609290386, 2.81052550495562783546795003995, 2.86055837631765748405711725966, 3.10764907866828550411708338943, 3.37839948092828449855903856871, 3.41603409681711236840952923078, 3.43395750938636894574526573694, 3.46454143679001584635421554695, 3.99378935399928321483722585588, 4.17711065949868828021311528243, 4.22482062425475065954444429449, 4.23425116868559581170039884697, 4.45228163340157235125809958289, 4.77423615118698125310883709925, 4.95663466285069125706354709223, 5.05229496815414267386762718627

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

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