Properties

Label 12-690e6-1.1-c1e6-0-0
Degree $12$
Conductor $1.079\times 10^{17}$
Sign $1$
Analytic cond. $27974.1$
Root an. cond. $2.34727$
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  − 3·4-s − 3·9-s + 6·16-s + 8·19-s + 25-s − 28·29-s − 24·31-s + 9·36-s − 12·41-s + 14·49-s − 4·59-s − 28·61-s − 10·64-s + 12·71-s − 24·76-s + 8·79-s + 6·81-s − 16·89-s − 3·100-s + 44·101-s + 12·109-s + 84·116-s − 34·121-s + 72·124-s + 16·125-s + 127-s + 131-s + ⋯
L(s)  = 1  − 3/2·4-s − 9-s + 3/2·16-s + 1.83·19-s + 1/5·25-s − 5.19·29-s − 4.31·31-s + 3/2·36-s − 1.87·41-s + 2·49-s − 0.520·59-s − 3.58·61-s − 5/4·64-s + 1.42·71-s − 2.75·76-s + 0.900·79-s + 2/3·81-s − 1.69·89-s − 0.299·100-s + 4.37·101-s + 1.14·109-s + 7.79·116-s − 3.09·121-s + 6.46·124-s + 1.43·125-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 23^{6}\)
Sign: $1$
Analytic conductor: \(27974.1\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{690} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 23^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5646329139\)
\(L(\frac12)\) \(\approx\) \(0.5646329139\)
\(L(\frac{3}{2})\) 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 \( ( 1 + T^{2} )^{3} \)
5 \( 1 - T^{2} - 16 T^{3} - p T^{4} + p^{3} T^{6} \)
23 \( ( 1 + T^{2} )^{3} \)
good7 \( 1 - 2 p T^{2} + 127 T^{4} - 1028 T^{6} + 127 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \)
11 \( ( 1 + 17 T^{2} - 16 T^{3} + 17 p T^{4} + p^{3} T^{6} )^{2} \)
13 \( 1 - 46 T^{2} + 1063 T^{4} - 16228 T^{6} + 1063 p^{2} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 58 T^{2} + 1455 T^{4} - 25708 T^{6} + 1455 p^{2} T^{8} - 58 p^{4} T^{10} + p^{6} T^{12} \)
19 \( ( 1 - 4 T + 49 T^{2} - 136 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( ( 1 + 14 T + 91 T^{2} + 468 T^{3} + 91 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( ( 1 + 4 T + p T^{2} )^{6} \)
37 \( 1 - 18 T^{2} - 265 T^{4} + 21412 T^{6} - 265 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \)
41 \( ( 1 + 6 T + 71 T^{2} + 500 T^{3} + 71 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( 1 - 150 T^{2} + 9335 T^{4} - 407060 T^{6} + 9335 p^{2} T^{8} - 150 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 - 90 T^{2} + 5231 T^{4} - 236204 T^{6} + 5231 p^{2} T^{8} - 90 p^{4} T^{10} + p^{6} T^{12} \)
53 \( 1 - 258 T^{2} + 29975 T^{4} - 2025596 T^{6} + 29975 p^{2} T^{8} - 258 p^{4} T^{10} + p^{6} T^{12} \)
59 \( ( 1 + 2 T + 93 T^{2} + 132 T^{3} + 93 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 + 14 T + 115 T^{2} + 596 T^{3} + 115 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 198 T^{2} + 22055 T^{4} - 1725428 T^{6} + 22055 p^{2} T^{8} - 198 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 - 6 T + 89 T^{2} - 1084 T^{3} + 89 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 310 T^{2} + 45631 T^{4} - 4119796 T^{6} + 45631 p^{2} T^{8} - 310 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 - 4 T + 93 T^{2} - 40 T^{3} + 93 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 322 T^{2} + 51047 T^{4} - 5155260 T^{6} + 51047 p^{2} T^{8} - 322 p^{4} T^{10} + p^{6} T^{12} \)
89 \( ( 1 + 8 T + 235 T^{2} + 1296 T^{3} + 235 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 330 T^{2} + 57999 T^{4} - 6805708 T^{6} + 57999 p^{2} T^{8} - 330 p^{4} T^{10} + p^{6} T^{12} \)
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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

−5.49288341192927043017928742824, −5.44541610131936594297340118509, −5.43706490544826856756977452236, −5.11996696825022873012102947786, −4.94530519209386311746993164876, −4.79602761902937948653697760210, −4.67200353072705918438445387483, −4.42056191506382165798247338144, −4.01822819301975809525106485614, −3.97279167040471620375744686712, −3.80224340879444563616649041099, −3.70957367324750481010698043165, −3.36444271991513187527168419979, −3.32651984691583597331848678487, −3.31249983328288677068321483242, −3.04720196881521653527097991728, −2.71571553421871008147453122111, −2.20330752512858384749545525974, −2.09291973819645089464573489828, −1.99778208585711164553479391031, −1.57560809319713457757397663122, −1.57517351925560974559097061736, −1.10442929055303463146516145853, −0.36468855572919945059992450275, −0.31488607278915811316398700563, 0.31488607278915811316398700563, 0.36468855572919945059992450275, 1.10442929055303463146516145853, 1.57517351925560974559097061736, 1.57560809319713457757397663122, 1.99778208585711164553479391031, 2.09291973819645089464573489828, 2.20330752512858384749545525974, 2.71571553421871008147453122111, 3.04720196881521653527097991728, 3.31249983328288677068321483242, 3.32651984691583597331848678487, 3.36444271991513187527168419979, 3.70957367324750481010698043165, 3.80224340879444563616649041099, 3.97279167040471620375744686712, 4.01822819301975809525106485614, 4.42056191506382165798247338144, 4.67200353072705918438445387483, 4.79602761902937948653697760210, 4.94530519209386311746993164876, 5.11996696825022873012102947786, 5.43706490544826856756977452236, 5.44541610131936594297340118509, 5.49288341192927043017928742824

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

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