Properties

Label 12-912e6-1.1-c1e6-0-3
Degree $12$
Conductor $5.754\times 10^{17}$
Sign $1$
Analytic cond. $149152.$
Root an. cond. $2.69858$
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  − 15·13-s − 3·19-s + 9·27-s − 39·43-s + 42·61-s − 33·67-s − 51·73-s + 12·79-s + 33·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 111·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 4.16·13-s − 0.688·19-s + 1.73·27-s − 5.94·43-s + 5.37·61-s − 4.03·67-s − 5.96·73-s + 1.35·79-s + 3·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(149152.\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{912} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.4926020150\)
\(L(\frac12)\) \(\approx\) \(0.4926020150\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - p^{2} T^{3} + p^{3} T^{6} \)
19 \( ( 1 + T + p T^{2} )^{3} \)
good5 \( 1 + p^{3} T^{6} + p^{6} T^{12} \)
7 \( ( 1 - 17 T^{3} + p^{3} T^{6} )( 1 + 17 T^{3} + p^{3} T^{6} ) \)
11 \( ( 1 - p T^{2} + p^{2} T^{4} )^{3} \)
13 \( ( 1 + 5 T + p T^{2} )^{3}( 1 + 89 T^{3} + p^{3} T^{6} ) \)
17 \( 1 + p^{3} T^{6} + p^{6} T^{12} \)
23 \( 1 - p^{3} T^{6} + p^{6} T^{12} \)
29 \( 1 + p^{3} T^{6} + p^{6} T^{12} \)
31 \( ( 1 - 19 T^{3} + p^{3} T^{6} )( 1 + 19 T^{3} + p^{3} T^{6} ) \)
37 \( ( 1 - 323 T^{3} + p^{3} T^{6} )^{2} \)
41 \( 1 + p^{3} T^{6} + p^{6} T^{12} \)
43 \( ( 1 + 13 T + p T^{2} )^{3}( 1 + 449 T^{3} + p^{3} T^{6} ) \)
47 \( 1 - p^{3} T^{6} + p^{6} T^{12} \)
53 \( 1 + p^{3} T^{6} + p^{6} T^{12} \)
59 \( 1 - p^{3} T^{6} + p^{6} T^{12} \)
61 \( ( 1 - 14 T + p T^{2} )^{3}( 1 + 901 T^{3} + p^{3} T^{6} ) \)
67 \( ( 1 + 11 T + p T^{2} )^{3}( 1 + 127 T^{3} + p^{3} T^{6} ) \)
71 \( 1 - p^{3} T^{6} + p^{6} T^{12} \)
73 \( ( 1 + 17 T + p T^{2} )^{3}( 1 - 919 T^{3} + p^{3} T^{6} ) \)
79 \( ( 1 - 4 T + p T^{2} )^{3}( 1 - 503 T^{3} + p^{3} T^{6} ) \)
83 \( ( 1 - p T^{2} + p^{2} T^{4} )^{3} \)
89 \( 1 + p^{3} T^{6} + p^{6} T^{12} \)
97 \( ( 1 - 523 T^{3} + p^{3} T^{6} )( 1 + 1853 T^{3} + p^{3} T^{6} ) \)
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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

−5.25634109765324018923881494419, −5.16335961627371743149034889173, −5.07831340279622912027278902826, −4.68517803620352043427949843448, −4.65889977553599189344570510687, −4.59898917768336375676170809003, −4.59118378753662316257461448643, −4.47319995445359238366800532126, −3.88773978048736782152851770130, −3.87233968586282071726460677139, −3.59494218507670877212415666207, −3.47886478186583444576426132453, −3.29135045865570922281486033021, −2.91336709552213079249836290300, −2.86848151282677363848587909329, −2.63260837751531576867232137554, −2.60446840063220201743174165940, −2.37460225858144084540206347908, −1.96513582016621737070182723937, −1.88871243830551234459117170696, −1.70865869402780470008787520636, −1.23413651455424768228571368041, −1.19408425453849615831208732409, −0.32589764396672231637462322538, −0.22880649303927208142509817992, 0.22880649303927208142509817992, 0.32589764396672231637462322538, 1.19408425453849615831208732409, 1.23413651455424768228571368041, 1.70865869402780470008787520636, 1.88871243830551234459117170696, 1.96513582016621737070182723937, 2.37460225858144084540206347908, 2.60446840063220201743174165940, 2.63260837751531576867232137554, 2.86848151282677363848587909329, 2.91336709552213079249836290300, 3.29135045865570922281486033021, 3.47886478186583444576426132453, 3.59494218507670877212415666207, 3.87233968586282071726460677139, 3.88773978048736782152851770130, 4.47319995445359238366800532126, 4.59118378753662316257461448643, 4.59898917768336375676170809003, 4.65889977553599189344570510687, 4.68517803620352043427949843448, 5.07831340279622912027278902826, 5.16335961627371743149034889173, 5.25634109765324018923881494419

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

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