Properties

Label 12-912e6-1.1-c1e6-0-10
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 21·13-s − 21·19-s − 9·27-s + 15·43-s + 42·61-s − 15·67-s + 21·73-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 + 255·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 5.82·13-s − 4.81·19-s − 1.73·27-s + 2.28·43-s + 5.37·61-s − 1.83·67-s + 2.45·73-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 + 19.6·169-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

\(L(1)\) \(\approx\) \(5.257840116\)
\(L(\frac12)\) \(\approx\) \(5.257840116\)
\(L(\frac{3}{2})\) 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 \( ( 1 + 7 T + p T^{2} )^{3} \)
good5 \( 1 + p^{3} T^{6} + p^{6} T^{12} \)
7 \( ( 1 - 37 T^{3} + p^{3} T^{6} )( 1 + 37 T^{3} + p^{3} T^{6} ) \)
11 \( ( 1 - p T^{2} + p^{2} T^{4} )^{3} \)
13 \( ( 1 - 7 T + p T^{2} )^{3}( 1 - 19 T^{3} + p^{3} T^{6} ) \)
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 \( ( 1 - 289 T^{3} + p^{3} T^{6} )( 1 + 289 T^{3} + p^{3} T^{6} ) \)
37 \( ( 1 + 433 T^{3} + p^{3} T^{6} )^{2} \)
41 \( 1 + p^{3} T^{6} + p^{6} T^{12} \)
43 \( ( 1 - 5 T + p T^{2} )^{3}( 1 + 71 T^{3} + p^{3} T^{6} ) \)
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 \( ( 1 - 14 T + p T^{2} )^{3}( 1 - 719 T^{3} + p^{3} T^{6} ) \)
67 \( ( 1 + 5 T + p T^{2} )^{3}( 1 - 1007 T^{3} + p^{3} T^{6} ) \)
71 \( 1 - p^{3} T^{6} + p^{6} T^{12} \)
73 \( ( 1 - 7 T + p T^{2} )^{3}( 1 - 271 T^{3} + p^{3} T^{6} ) \)
79 \( ( 1 - 4 T + p T^{2} )^{3}( 1 + 1387 T^{3} + p^{3} T^{6} ) \)
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

−5.60058199761723356052387185821, −5.34650427774281322646115938425, −4.93034679147730976527107135942, −4.79645711889459436043852547517, −4.61173064175108512620433479638, −4.39196480032787989397657985610, −4.23318282450304882464179375481, −4.13265430857329472101969863962, −3.99042330144288607300727314928, −3.91818425591858101935111919367, −3.65576373657411379668787112259, −3.51298642229224985707121537180, −3.42128709579017040022305832584, −3.28717168848648056933164034536, −3.03667455372289467366538331515, −2.53418342428528841579776235084, −2.30354075601540642407451647942, −2.18531789137901414025106994488, −2.09152640630741846394514312315, −1.83419764776684334933185900845, −1.72785062441568952082611574164, −1.17448830980954796030095028679, −0.939596367726316550875704512532, −0.878673420246117483968296098689, −0.38463006401446620796118371260, 0.38463006401446620796118371260, 0.878673420246117483968296098689, 0.939596367726316550875704512532, 1.17448830980954796030095028679, 1.72785062441568952082611574164, 1.83419764776684334933185900845, 2.09152640630741846394514312315, 2.18531789137901414025106994488, 2.30354075601540642407451647942, 2.53418342428528841579776235084, 3.03667455372289467366538331515, 3.28717168848648056933164034536, 3.42128709579017040022305832584, 3.51298642229224985707121537180, 3.65576373657411379668787112259, 3.91818425591858101935111919367, 3.99042330144288607300727314928, 4.13265430857329472101969863962, 4.23318282450304882464179375481, 4.39196480032787989397657985610, 4.61173064175108512620433479638, 4.79645711889459436043852547517, 4.93034679147730976527107135942, 5.34650427774281322646115938425, 5.60058199761723356052387185821

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

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