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  − 1.13·2-s + 3-s − 0.703·4-s + 2.27·5-s − 1.13·6-s + 3.07·8-s + 9-s − 2.58·10-s + 4.11·11-s − 0.703·12-s − 1.02·13-s + 2.27·15-s − 2.09·16-s + 1.01·17-s − 1.13·18-s + 0.361·19-s − 1.59·20-s − 4.68·22-s − 4.61·23-s + 3.07·24-s + 0.158·25-s + 1.16·26-s + 27-s − 1.50·29-s − 2.58·30-s + 3.87·31-s − 3.76·32-s + ⋯
L(s)  = 1  − 0.805·2-s + 0.577·3-s − 0.351·4-s + 1.01·5-s − 0.464·6-s + 1.08·8-s + 0.333·9-s − 0.817·10-s + 1.24·11-s − 0.203·12-s − 0.283·13-s + 0.586·15-s − 0.524·16-s + 0.247·17-s − 0.268·18-s + 0.0828·19-s − 0.357·20-s − 0.998·22-s − 0.963·23-s + 0.628·24-s + 0.0316·25-s + 0.228·26-s + 0.192·27-s − 0.279·29-s − 0.472·30-s + 0.695·31-s − 0.666·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.958189225$
$L(\frac12)$  $\approx$  $1.958189225$
$L(\frac{3}{2})$   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 \( 1 + 1.13T + 2T^{2} \)
5 \( 1 - 2.27T + 5T^{2} \)
11 \( 1 - 4.11T + 11T^{2} \)
13 \( 1 + 1.02T + 13T^{2} \)
17 \( 1 - 1.01T + 17T^{2} \)
19 \( 1 - 0.361T + 19T^{2} \)
23 \( 1 + 4.61T + 23T^{2} \)
29 \( 1 + 1.50T + 29T^{2} \)
31 \( 1 - 3.87T + 31T^{2} \)
37 \( 1 - 9.23T + 37T^{2} \)
43 \( 1 + 2.19T + 43T^{2} \)
47 \( 1 + 1.32T + 47T^{2} \)
53 \( 1 - 7.53T + 53T^{2} \)
59 \( 1 + 1.63T + 59T^{2} \)
61 \( 1 - 14.8T + 61T^{2} \)
67 \( 1 - 5.38T + 67T^{2} \)
71 \( 1 + 1.61T + 71T^{2} \)
73 \( 1 - 2.22T + 73T^{2} \)
79 \( 1 + 4.15T + 79T^{2} \)
83 \( 1 - 1.53T + 83T^{2} \)
89 \( 1 - 1.06T + 89T^{2} \)
97 \( 1 - 0.418T + 97T^{2} \)
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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

−8.203347744111307587928264682360, −7.60139260760279471673095984480, −6.75578694222377611236277753490, −6.07177042741860214300068130353, −5.23950293653190218624754990631, −4.29943427504125778110649571558, −3.72978984463416845581511236531, −2.47844486287393200055789579772, −1.72967010409129227262335195479, −0.862959899299200933696688549984, 0.862959899299200933696688549984, 1.72967010409129227262335195479, 2.47844486287393200055789579772, 3.72978984463416845581511236531, 4.29943427504125778110649571558, 5.23950293653190218624754990631, 6.07177042741860214300068130353, 6.75578694222377611236277753490, 7.60139260760279471673095984480, 8.203347744111307587928264682360

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