Properties

Degree 12
Conductor $ 7^{12} \cdot 41^{6} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 6-s + 9-s − 12-s + 2·13-s − 17-s + 18-s − 19-s + 23-s − 3·25-s + 2·26-s − 34-s + 36-s + 37-s − 38-s − 2·39-s − 6·41-s − 2·43-s + 46-s − 47-s − 3·50-s + 51-s + 2·52-s + 57-s − 68-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − 6-s + 9-s − 12-s + 2·13-s − 17-s + 18-s − 19-s + 23-s − 3·25-s + 2·26-s − 34-s + 36-s + 37-s − 38-s − 2·39-s − 6·41-s − 2·43-s + 46-s − 47-s − 3·50-s + 51-s + 2·52-s + 57-s − 68-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 41^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 41^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(12\)
\( N \)  =  \(7^{12} \cdot 41^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{2009} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((12,\ 7^{12} \cdot 41^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.3580768895\)
\(L(\frac12)\)  \(\approx\)  \(0.3580768895\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{7,\;41\}$,\(F_p(T)\) is a polynomial of degree 12. If $p \in \{7,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 11.
$p$$F_p(T)$
bad7 \( 1 \)
41 \( ( 1 + T )^{6} \)
good2 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
3 \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
5 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
11 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
13 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
17 \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
19 \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
23 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
29 \( ( 1 - T )^{6}( 1 + T )^{6} \)
31 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
37 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
43 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
47 \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
53 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
59 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
61 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
67 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
71 \( ( 1 - T )^{6}( 1 + T )^{6} \)
73 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
79 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
83 \( ( 1 - T )^{6}( 1 + T )^{6} \)
89 \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−5.04401577427435506798130622400, −4.82836953762482163783993782437, −4.63375378762089689495879156420, −4.55516777346898850179205652688, −4.50092349371456262817860635579, −4.24639969694681809920604621951, −3.93556652893087677963129953892, −3.90542753497502932814445841013, −3.87345243392989194229338668329, −3.60603968135986095328287220566, −3.54652072770348459017211618959, −3.24868771091844489836162223007, −3.15988414154023773290477849106, −3.07571694836446014133026789368, −2.96112448191744686442098279712, −2.53557174314562538911207541897, −2.38895967264862872838212690630, −2.13557826763703809556282800369, −2.03449118281866938440195291613, −1.76580778284001710353102003885, −1.65919010299767152506857707676, −1.39791643264655361206048294006, −1.28325315793794628384740804361, −1.03645408969807067672409411965, −0.20020009897710312996831977629, 0.20020009897710312996831977629, 1.03645408969807067672409411965, 1.28325315793794628384740804361, 1.39791643264655361206048294006, 1.65919010299767152506857707676, 1.76580778284001710353102003885, 2.03449118281866938440195291613, 2.13557826763703809556282800369, 2.38895967264862872838212690630, 2.53557174314562538911207541897, 2.96112448191744686442098279712, 3.07571694836446014133026789368, 3.15988414154023773290477849106, 3.24868771091844489836162223007, 3.54652072770348459017211618959, 3.60603968135986095328287220566, 3.87345243392989194229338668329, 3.90542753497502932814445841013, 3.93556652893087677963129953892, 4.24639969694681809920604621951, 4.50092349371456262817860635579, 4.55516777346898850179205652688, 4.63375378762089689495879156420, 4.82836953762482163783993782437, 5.04401577427435506798130622400

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

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