Properties

Degree 2
Conductor $ 3 \cdot 7^{2} \cdot 41 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s − 3.46·5-s + 9-s + 2.26·11-s − 2·12-s + 1.46·13-s − 3.46·15-s + 4·16-s − 5·17-s − 7.46·19-s + 6.92·20-s − 1.46·23-s + 6.99·25-s + 27-s + 5.19·29-s − 2.26·31-s + 2.26·33-s − 2·36-s − 37-s + 1.46·39-s + 41-s − 3.92·43-s − 4.53·44-s − 3.46·45-s − 1.92·47-s + 4·48-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 1.54·5-s + 0.333·9-s + 0.683·11-s − 0.577·12-s + 0.406·13-s − 0.894·15-s + 16-s − 1.21·17-s − 1.71·19-s + 1.54·20-s − 0.305·23-s + 1.39·25-s + 0.192·27-s + 0.964·29-s − 0.407·31-s + 0.394·33-s − 0.333·36-s − 0.164·37-s + 0.234·39-s + 0.156·41-s − 0.599·43-s − 0.683·44-s − 0.516·45-s − 0.281·47-s + 0.577·48-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6027} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8320492335$
$L(\frac12)$  $\approx$  $0.8320492335$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;41\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 \)
41 \( 1 - T \)
good2 \( 1 + 2T^{2} \)
5 \( 1 + 3.46T + 5T^{2} \)
11 \( 1 - 2.26T + 11T^{2} \)
13 \( 1 - 1.46T + 13T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
19 \( 1 + 7.46T + 19T^{2} \)
23 \( 1 + 1.46T + 23T^{2} \)
29 \( 1 - 5.19T + 29T^{2} \)
31 \( 1 + 2.26T + 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
43 \( 1 + 3.92T + 43T^{2} \)
47 \( 1 + 1.92T + 47T^{2} \)
53 \( 1 + 2.92T + 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 + 1.73T + 61T^{2} \)
67 \( 1 - 0.535T + 67T^{2} \)
71 \( 1 - 2.80T + 71T^{2} \)
73 \( 1 + 1.19T + 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 + 10T + 83T^{2} \)
89 \( 1 - 4.92T + 89T^{2} \)
97 \( 1 - 18.3T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.332084222414555977303129528652, −7.55720524177333256665773263579, −6.76046695167375716021199324542, −6.07985993550982072504098221293, −4.70148630756669300876138364389, −4.37617718652479557286822383532, −3.80748329693678630939012360002, −3.08415766596524700145753616135, −1.80093794891218194311150207312, −0.46538309134254912408897046436, 0.46538309134254912408897046436, 1.80093794891218194311150207312, 3.08415766596524700145753616135, 3.80748329693678630939012360002, 4.37617718652479557286822383532, 4.70148630756669300876138364389, 6.07985993550982072504098221293, 6.76046695167375716021199324542, 7.55720524177333256665773263579, 8.332084222414555977303129528652

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