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  + 1.33·2-s + 3-s − 0.230·4-s − 3.44·5-s + 1.33·6-s − 2.96·8-s + 9-s − 4.58·10-s + 2.42·11-s − 0.230·12-s − 4.50·13-s − 3.44·15-s − 3.48·16-s − 4.56·17-s + 1.33·18-s − 1.29·19-s + 0.794·20-s + 3.23·22-s − 5.25·23-s − 2.96·24-s + 6.85·25-s − 5.99·26-s + 27-s + 6.83·29-s − 4.58·30-s + 2.39·31-s + 1.29·32-s + ⋯
L(s)  = 1  + 0.940·2-s + 0.577·3-s − 0.115·4-s − 1.54·5-s + 0.543·6-s − 1.04·8-s + 0.333·9-s − 1.44·10-s + 0.732·11-s − 0.0666·12-s − 1.25·13-s − 0.889·15-s − 0.871·16-s − 1.10·17-s + 0.313·18-s − 0.296·19-s + 0.177·20-s + 0.689·22-s − 1.09·23-s − 0.605·24-s + 1.37·25-s − 1.17·26-s + 0.192·27-s + 1.26·29-s − 0.836·30-s + 0.429·31-s + 0.229·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.642641623$
$L(\frac12)$  $\approx$  $1.642641623$
$L(\frac{3}{2})$   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 \( 1 - 1.33T + 2T^{2} \)
5 \( 1 + 3.44T + 5T^{2} \)
11 \( 1 - 2.42T + 11T^{2} \)
13 \( 1 + 4.50T + 13T^{2} \)
17 \( 1 + 4.56T + 17T^{2} \)
19 \( 1 + 1.29T + 19T^{2} \)
23 \( 1 + 5.25T + 23T^{2} \)
29 \( 1 - 6.83T + 29T^{2} \)
31 \( 1 - 2.39T + 31T^{2} \)
37 \( 1 + 0.323T + 37T^{2} \)
43 \( 1 - 3.93T + 43T^{2} \)
47 \( 1 - 12.2T + 47T^{2} \)
53 \( 1 - 5.17T + 53T^{2} \)
59 \( 1 + 1.20T + 59T^{2} \)
61 \( 1 + 4.84T + 61T^{2} \)
67 \( 1 - 3.77T + 67T^{2} \)
71 \( 1 + 5.00T + 71T^{2} \)
73 \( 1 - 8.20T + 73T^{2} \)
79 \( 1 + 7.91T + 79T^{2} \)
83 \( 1 - 2.47T + 83T^{2} \)
89 \( 1 + 2.30T + 89T^{2} \)
97 \( 1 + 15.4T + 97T^{2} \)
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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

−8.132861427129449719187479359799, −7.29216842035512215068102692877, −6.74907014317184691428664969991, −5.87964609505640546297964792941, −4.76078862100431622750585382826, −4.31806131743470649544938659634, −3.91867600034266714777262085941, −3.01266911009208819175453707315, −2.27364198583883526971081816240, −0.54264510288037499532358963411, 0.54264510288037499532358963411, 2.27364198583883526971081816240, 3.01266911009208819175453707315, 3.91867600034266714777262085941, 4.31806131743470649544938659634, 4.76078862100431622750585382826, 5.87964609505640546297964792941, 6.74907014317184691428664969991, 7.29216842035512215068102692877, 8.132861427129449719187479359799

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