Properties

Degree 2
Conductor $ 3 \cdot 7^{2} \cdot 41 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.766·2-s + 3-s − 1.41·4-s + 0.561·5-s − 0.766·6-s + 2.61·8-s + 9-s − 0.430·10-s − 3.11·11-s − 1.41·12-s + 5.92·13-s + 0.561·15-s + 0.823·16-s + 3.15·17-s − 0.766·18-s − 0.142·19-s − 0.793·20-s + 2.38·22-s − 4.62·23-s + 2.61·24-s − 4.68·25-s − 4.53·26-s + 27-s − 8.30·29-s − 0.430·30-s − 0.336·31-s − 5.86·32-s + ⋯
L(s)  = 1  − 0.541·2-s + 0.577·3-s − 0.706·4-s + 0.251·5-s − 0.312·6-s + 0.924·8-s + 0.333·9-s − 0.136·10-s − 0.938·11-s − 0.407·12-s + 1.64·13-s + 0.145·15-s + 0.205·16-s + 0.764·17-s − 0.180·18-s − 0.0326·19-s − 0.177·20-s + 0.508·22-s − 0.965·23-s + 0.533·24-s − 0.936·25-s − 0.890·26-s + 0.192·27-s − 1.54·29-s − 0.0785·30-s − 0.0604·31-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  =  1
Selberg data  =  $(2,\ 6027,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 0.766T + 2T^{2} \)
5 \( 1 - 0.561T + 5T^{2} \)
11 \( 1 + 3.11T + 11T^{2} \)
13 \( 1 - 5.92T + 13T^{2} \)
17 \( 1 - 3.15T + 17T^{2} \)
19 \( 1 + 0.142T + 19T^{2} \)
23 \( 1 + 4.62T + 23T^{2} \)
29 \( 1 + 8.30T + 29T^{2} \)
31 \( 1 + 0.336T + 31T^{2} \)
37 \( 1 + 9.44T + 37T^{2} \)
43 \( 1 - 9.28T + 43T^{2} \)
47 \( 1 + 7.35T + 47T^{2} \)
53 \( 1 + 8.42T + 53T^{2} \)
59 \( 1 - 0.911T + 59T^{2} \)
61 \( 1 + 3.48T + 61T^{2} \)
67 \( 1 + 0.537T + 67T^{2} \)
71 \( 1 + 0.835T + 71T^{2} \)
73 \( 1 - 8.54T + 73T^{2} \)
79 \( 1 + 10.4T + 79T^{2} \)
83 \( 1 + 5.99T + 83T^{2} \)
89 \( 1 - 18.7T + 89T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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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

−7.81913703836580628407827056002, −7.46235936974584055248380322451, −6.19740097369008641829403652728, −5.63106457784084258262873770745, −4.83005896415358642133314206341, −3.77961256319912985355771299643, −3.46257277494936833167249644042, −2.09032167829964142852757543338, −1.34859254131930283217143271422, 0, 1.34859254131930283217143271422, 2.09032167829964142852757543338, 3.46257277494936833167249644042, 3.77961256319912985355771299643, 4.83005896415358642133314206341, 5.63106457784084258262873770745, 6.19740097369008641829403652728, 7.46235936974584055248380322451, 7.81913703836580628407827056002

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