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  + 2.37·2-s − 3-s + 3.63·4-s + 3.47·5-s − 2.37·6-s + 3.87·8-s + 9-s + 8.24·10-s + 1.00·11-s − 3.63·12-s + 3.69·13-s − 3.47·15-s + 1.93·16-s − 2.26·17-s + 2.37·18-s − 3.08·19-s + 12.6·20-s + 2.39·22-s + 2.97·23-s − 3.87·24-s + 7.07·25-s + 8.77·26-s − 27-s + 6.10·29-s − 8.24·30-s − 1.10·31-s − 3.15·32-s + ⋯
L(s)  = 1  + 1.67·2-s − 0.577·3-s + 1.81·4-s + 1.55·5-s − 0.969·6-s + 1.37·8-s + 0.333·9-s + 2.60·10-s + 0.304·11-s − 1.04·12-s + 1.02·13-s − 0.897·15-s + 0.484·16-s − 0.549·17-s + 0.559·18-s − 0.707·19-s + 2.82·20-s + 0.510·22-s + 0.619·23-s − 0.791·24-s + 1.41·25-s + 1.72·26-s − 0.192·27-s + 1.13·29-s − 1.50·30-s − 0.197·31-s − 0.557·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.792105209$
$L(\frac12)$  $\approx$  $6.792105209$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
7 \( 1 \)
41 \( 1 - T \)
good2 \( 1 - 2.37T + 2T^{2} \)
5 \( 1 - 3.47T + 5T^{2} \)
11 \( 1 - 1.00T + 11T^{2} \)
13 \( 1 - 3.69T + 13T^{2} \)
17 \( 1 + 2.26T + 17T^{2} \)
19 \( 1 + 3.08T + 19T^{2} \)
23 \( 1 - 2.97T + 23T^{2} \)
29 \( 1 - 6.10T + 29T^{2} \)
31 \( 1 + 1.10T + 31T^{2} \)
37 \( 1 - 7.80T + 37T^{2} \)
43 \( 1 - 11.3T + 43T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 + 3.15T + 53T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 + 2.86T + 61T^{2} \)
67 \( 1 + 7.54T + 67T^{2} \)
71 \( 1 - 7.56T + 71T^{2} \)
73 \( 1 - 8.49T + 73T^{2} \)
79 \( 1 - 8.24T + 79T^{2} \)
83 \( 1 - 12.4T + 83T^{2} \)
89 \( 1 - 7.10T + 89T^{2} \)
97 \( 1 + 1.37T + 97T^{2} \)
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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

−7.83296401263847885310902311085, −6.71815545949181507946422926929, −6.20659564488397028707730197202, −6.11931799164458775017348742978, −5.13815800261778791782317818576, −4.65402190428188787196580071332, −3.84002718063870328831134852586, −2.85515393312265068904819477270, −2.11764368047744518963061809073, −1.19814100491402109706838340286, 1.19814100491402109706838340286, 2.11764368047744518963061809073, 2.85515393312265068904819477270, 3.84002718063870328831134852586, 4.65402190428188787196580071332, 5.13815800261778791782317818576, 6.11931799164458775017348742978, 6.20659564488397028707730197202, 6.71815545949181507946422926929, 7.83296401263847885310902311085

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