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  − 2.23·2-s + 3-s + 2.97·4-s − 0.893·5-s − 2.23·6-s − 2.18·8-s + 9-s + 1.99·10-s − 3.03·11-s + 2.97·12-s − 1.02·13-s − 0.893·15-s − 1.08·16-s − 0.590·17-s − 2.23·18-s − 3.50·19-s − 2.65·20-s + 6.76·22-s − 0.556·23-s − 2.18·24-s − 4.20·25-s + 2.28·26-s + 27-s + 7.89·29-s + 1.99·30-s + 1.28·31-s + 6.79·32-s + ⋯
L(s)  = 1  − 1.57·2-s + 0.577·3-s + 1.48·4-s − 0.399·5-s − 0.910·6-s − 0.771·8-s + 0.333·9-s + 0.630·10-s − 0.914·11-s + 0.859·12-s − 0.283·13-s − 0.230·15-s − 0.271·16-s − 0.143·17-s − 0.525·18-s − 0.804·19-s − 0.594·20-s + 1.44·22-s − 0.115·23-s − 0.445·24-s − 0.840·25-s + 0.447·26-s + 0.192·27-s + 1.46·29-s + 0.363·30-s + 0.230·31-s + 1.20·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6051235739$
$L(\frac12)$  $\approx$  $0.6051235739$
$L(\frac{3}{2})$   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 \( 1 + 2.23T + 2T^{2} \)
5 \( 1 + 0.893T + 5T^{2} \)
11 \( 1 + 3.03T + 11T^{2} \)
13 \( 1 + 1.02T + 13T^{2} \)
17 \( 1 + 0.590T + 17T^{2} \)
19 \( 1 + 3.50T + 19T^{2} \)
23 \( 1 + 0.556T + 23T^{2} \)
29 \( 1 - 7.89T + 29T^{2} \)
31 \( 1 - 1.28T + 31T^{2} \)
37 \( 1 + 11.5T + 37T^{2} \)
43 \( 1 + 5.76T + 43T^{2} \)
47 \( 1 - 0.229T + 47T^{2} \)
53 \( 1 - 8.54T + 53T^{2} \)
59 \( 1 + 10.6T + 59T^{2} \)
61 \( 1 + 8.09T + 61T^{2} \)
67 \( 1 - 8.13T + 67T^{2} \)
71 \( 1 - 1.38T + 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 - 4.67T + 79T^{2} \)
83 \( 1 - 3.22T + 83T^{2} \)
89 \( 1 - 15.9T + 89T^{2} \)
97 \( 1 - 5.16T + 97T^{2} \)
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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

−8.102724421406290057384581623835, −7.77003466887083646812306209009, −6.94399492469264022381223161775, −6.38876923904216168066650506039, −5.17862299948950945590114925318, −4.39826629692497276432221219142, −3.36713806089681207927445296299, −2.44328963937335634681052745926, −1.77832969501879149481219902759, −0.48973628615642097886555838330, 0.48973628615642097886555838330, 1.77832969501879149481219902759, 2.44328963937335634681052745926, 3.36713806089681207927445296299, 4.39826629692497276432221219142, 5.17862299948950945590114925318, 6.38876923904216168066650506039, 6.94399492469264022381223161775, 7.77003466887083646812306209009, 8.102724421406290057384581623835

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