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  + 0.718·2-s − 3-s − 1.48·4-s + 3.61·5-s − 0.718·6-s − 2.50·8-s + 9-s + 2.60·10-s + 4.77·11-s + 1.48·12-s + 1.67·13-s − 3.61·15-s + 1.16·16-s − 4.10·17-s + 0.718·18-s − 7.21·19-s − 5.37·20-s + 3.42·22-s + 6.09·23-s + 2.50·24-s + 8.10·25-s + 1.20·26-s − 27-s + 4.27·29-s − 2.60·30-s + 9.68·31-s + 5.84·32-s + ⋯
L(s)  = 1  + 0.508·2-s − 0.577·3-s − 0.741·4-s + 1.61·5-s − 0.293·6-s − 0.884·8-s + 0.333·9-s + 0.822·10-s + 1.43·11-s + 0.428·12-s + 0.465·13-s − 0.934·15-s + 0.292·16-s − 0.996·17-s + 0.169·18-s − 1.65·19-s − 1.20·20-s + 0.730·22-s + 1.27·23-s + 0.510·24-s + 1.62·25-s + 0.236·26-s − 0.192·27-s + 0.794·29-s − 0.474·30-s + 1.74·31-s + 1.03·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.559575919$
$L(\frac12)$  $\approx$  $2.559575919$
$L(\frac{3}{2})$   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 \( 1 - 0.718T + 2T^{2} \)
5 \( 1 - 3.61T + 5T^{2} \)
11 \( 1 - 4.77T + 11T^{2} \)
13 \( 1 - 1.67T + 13T^{2} \)
17 \( 1 + 4.10T + 17T^{2} \)
19 \( 1 + 7.21T + 19T^{2} \)
23 \( 1 - 6.09T + 23T^{2} \)
29 \( 1 - 4.27T + 29T^{2} \)
31 \( 1 - 9.68T + 31T^{2} \)
37 \( 1 + 9.29T + 37T^{2} \)
43 \( 1 - 0.493T + 43T^{2} \)
47 \( 1 - 0.929T + 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 + 0.276T + 59T^{2} \)
61 \( 1 + 8.28T + 61T^{2} \)
67 \( 1 + 6.10T + 67T^{2} \)
71 \( 1 + 4.71T + 71T^{2} \)
73 \( 1 - 9.42T + 73T^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 - 8.22T + 83T^{2} \)
89 \( 1 + 5.51T + 89T^{2} \)
97 \( 1 + 4.56T + 97T^{2} \)
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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

−8.415383677285993827289812371167, −6.81301261738892487519826205010, −6.45518069265842137317932283195, −6.05014601849225775977900632894, −5.14158728693039193416333671036, −4.58859595459327083319683854333, −3.89806023409231347628043843507, −2.78536874216101350051918263477, −1.78921727718960751016433582635, −0.846621230616195102577836592232, 0.846621230616195102577836592232, 1.78921727718960751016433582635, 2.78536874216101350051918263477, 3.89806023409231347628043843507, 4.58859595459327083319683854333, 5.14158728693039193416333671036, 6.05014601849225775977900632894, 6.45518069265842137317932283195, 6.81301261738892487519826205010, 8.415383677285993827289812371167

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