Properties

Label 12-9680e6-1.1-c1e6-0-3
Degree $12$
Conductor $8.227\times 10^{23}$
Sign $1$
Analytic cond. $2.13262\times 10^{11}$
Root an. cond. $8.79176$
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  + 2·3-s − 6·5-s + 3·9-s − 12·15-s − 4·23-s + 21·25-s + 10·27-s − 28·31-s + 32·37-s − 18·45-s + 6·47-s − 13·49-s − 4·53-s − 12·59-s + 26·67-s − 8·69-s + 12·71-s + 42·75-s + 23·81-s + 10·89-s − 56·93-s + 44·97-s − 52·103-s + 64·111-s + 28·113-s + 24·115-s − 56·125-s + ⋯
L(s)  = 1  + 1.15·3-s − 2.68·5-s + 9-s − 3.09·15-s − 0.834·23-s + 21/5·25-s + 1.92·27-s − 5.02·31-s + 5.26·37-s − 2.68·45-s + 0.875·47-s − 1.85·49-s − 0.549·53-s − 1.56·59-s + 3.17·67-s − 0.963·69-s + 1.42·71-s + 4.84·75-s + 23/9·81-s + 1.05·89-s − 5.80·93-s + 4.46·97-s − 5.12·103-s + 6.07·111-s + 2.63·113-s + 2.23·115-s − 5.00·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 5^{6} \cdot 11^{12}\)
Sign: $1$
Analytic conductor: \(2.13262\times 10^{11}\)
Root analytic conductor: \(8.79176\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 5^{6} \cdot 11^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.796660503\)
\(L(\frac12)\) \(\approx\) \(6.796660503\)
\(L(\frac{3}{2})\) 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 \( ( 1 + T )^{6} \)
11 \( 1 \)
good3 \( ( 1 - T - p T^{3} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
7 \( 1 + 13 T^{2} + 158 T^{4} + 1117 T^{6} + 158 p^{2} T^{8} + 13 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 + 10 T^{2} + 503 T^{4} + 3292 T^{6} + 503 p^{2} T^{8} + 10 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 + 22 T^{2} + 495 T^{4} + 7028 T^{6} + 495 p^{2} T^{8} + 22 p^{4} T^{10} + p^{6} T^{12} \)
19 \( 1 + 10 T^{2} + 1079 T^{4} + 7132 T^{6} + 1079 p^{2} T^{8} + 10 p^{4} T^{10} + p^{6} T^{12} \)
23 \( ( 1 + 2 T + 25 T^{2} + 104 T^{3} + 25 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( 1 + 94 T^{2} + 4935 T^{4} + 169988 T^{6} + 4935 p^{2} T^{8} + 94 p^{4} T^{10} + p^{6} T^{12} \)
31 \( ( 1 + 14 T + 125 T^{2} + 848 T^{3} + 125 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( ( 1 - 16 T + 159 T^{2} - 1136 T^{3} + 159 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 133 T^{2} + 6846 T^{4} + 252041 T^{6} + 6846 p^{2} T^{8} + 133 p^{4} T^{10} + p^{6} T^{12} \)
43 \( 1 + 157 T^{2} + 12974 T^{4} + 688117 T^{6} + 12974 p^{2} T^{8} + 157 p^{4} T^{10} + p^{6} T^{12} \)
47 \( ( 1 - 3 T + 60 T^{2} - 201 T^{3} + 60 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
53 \( ( 1 + 2 T + 115 T^{2} + 224 T^{3} + 115 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( ( 1 + 6 T + 93 T^{2} + 168 T^{3} + 93 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( 1 + 277 T^{2} + 36422 T^{4} + 2817289 T^{6} + 36422 p^{2} T^{8} + 277 p^{4} T^{10} + p^{6} T^{12} \)
67 \( ( 1 - 13 T + 140 T^{2} - 847 T^{3} + 140 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
71 \( ( 1 - 6 T + 129 T^{2} - 312 T^{3} + 129 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{3} \)
79 \( 1 + 22 T^{2} - 721 T^{4} + 597364 T^{6} - 721 p^{2} T^{8} + 22 p^{4} T^{10} + p^{6} T^{12} \)
83 \( 1 + 154 T^{2} + 19095 T^{4} + 2118524 T^{6} + 19095 p^{2} T^{8} + 154 p^{4} T^{10} + p^{6} T^{12} \)
89 \( ( 1 - 5 T + 238 T^{2} - 833 T^{3} + 238 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( ( 1 - 22 T + 407 T^{2} - 4288 T^{3} + 407 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−3.87099585918686449259819795504, −3.84849562144555085602904400766, −3.50645014439262634346272842627, −3.49002918531380817584790200137, −3.35857587814011342807033188200, −3.32652862716918976347383333410, −3.31604781861549141416440760950, −3.01258550975391915385379459816, −2.76863931349823043566762662249, −2.74294197949376265760062130722, −2.56705231114062286865624056804, −2.48423734475476580047533570694, −2.31219607191845010274930464242, −2.01250659434059008165543651615, −1.98805207633491434456815011333, −1.97293154074098612475722838992, −1.59944006232141554947189814307, −1.52719097002605503647084665591, −1.38705860958705886151318579572, −1.02460905576222631865649010416, −0.898459652475036108815849831623, −0.65517986914311191993616089569, −0.55304061499723680554186985119, −0.45201600494989195267218535646, −0.21948547685300610240649924108, 0.21948547685300610240649924108, 0.45201600494989195267218535646, 0.55304061499723680554186985119, 0.65517986914311191993616089569, 0.898459652475036108815849831623, 1.02460905576222631865649010416, 1.38705860958705886151318579572, 1.52719097002605503647084665591, 1.59944006232141554947189814307, 1.97293154074098612475722838992, 1.98805207633491434456815011333, 2.01250659434059008165543651615, 2.31219607191845010274930464242, 2.48423734475476580047533570694, 2.56705231114062286865624056804, 2.74294197949376265760062130722, 2.76863931349823043566762662249, 3.01258550975391915385379459816, 3.31604781861549141416440760950, 3.32652862716918976347383333410, 3.35857587814011342807033188200, 3.49002918531380817584790200137, 3.50645014439262634346272842627, 3.84849562144555085602904400766, 3.87099585918686449259819795504

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

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