Properties

Label 12-912e6-1.1-c1e6-0-16
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

\(L(1)\) \(\approx\) \(15.77352034\)
\(L(\frac12)\) \(\approx\) \(15.77352034\)
\(L(\frac{3}{2})\) 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

−5.53270293939711316752333239865, −5.17605438608744266050150569044, −4.95911127261451179107371265787, −4.92364802181306403523263644456, −4.90872091441008293905082866199, −4.64924304910814006038268889469, −4.16099294160785799462110064314, −4.03110537253817268744364996477, −3.90005904600197433446640832305, −3.64697861429667417694791568072, −3.63932986761867062384658231802, −3.60837125631599962472601632515, −3.41410023476157409441800433146, −3.07349021045347893684887200869, −2.98564891590821772528208597074, −2.96650616139512915236529846683, −2.54318055572895655023706193626, −2.23489947164983164020652034546, −1.87419761118707398889274613373, −1.81299347664337585092066646460, −1.37039470674065887621776356869, −1.10754456864066126221804616678, −0.993292234175993067722742301365, −0.959360763431657525910158357600, −0.73935972759110011218159088906, 0.73935972759110011218159088906, 0.959360763431657525910158357600, 0.993292234175993067722742301365, 1.10754456864066126221804616678, 1.37039470674065887621776356869, 1.81299347664337585092066646460, 1.87419761118707398889274613373, 2.23489947164983164020652034546, 2.54318055572895655023706193626, 2.96650616139512915236529846683, 2.98564891590821772528208597074, 3.07349021045347893684887200869, 3.41410023476157409441800433146, 3.60837125631599962472601632515, 3.63932986761867062384658231802, 3.64697861429667417694791568072, 3.90005904600197433446640832305, 4.03110537253817268744364996477, 4.16099294160785799462110064314, 4.64924304910814006038268889469, 4.90872091441008293905082866199, 4.92364802181306403523263644456, 4.95911127261451179107371265787, 5.17605438608744266050150569044, 5.53270293939711316752333239865

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

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