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  − 1.64·2-s + 3-s + 0.692·4-s + 0.0817·5-s − 1.64·6-s + 2.14·8-s + 9-s − 0.134·10-s − 4.94·11-s + 0.692·12-s + 1.23·13-s + 0.0817·15-s − 4.90·16-s − 0.810·17-s − 1.64·18-s + 6.55·19-s + 0.0566·20-s + 8.11·22-s − 5.53·23-s + 2.14·24-s − 4.99·25-s − 2.02·26-s + 27-s − 0.0447·29-s − 0.134·30-s + 7.20·31-s + 3.75·32-s + ⋯
L(s)  = 1  − 1.16·2-s + 0.577·3-s + 0.346·4-s + 0.0365·5-s − 0.669·6-s + 0.758·8-s + 0.333·9-s − 0.0424·10-s − 1.49·11-s + 0.199·12-s + 0.341·13-s + 0.0211·15-s − 1.22·16-s − 0.196·17-s − 0.386·18-s + 1.50·19-s + 0.0126·20-s + 1.72·22-s − 1.15·23-s + 0.437·24-s − 0.998·25-s − 0.396·26-s + 0.192·27-s − 0.00830·29-s − 0.0244·30-s + 1.29·31-s + 0.664·32-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 \( 1 + 1.64T + 2T^{2} \)
5 \( 1 - 0.0817T + 5T^{2} \)
11 \( 1 + 4.94T + 11T^{2} \)
13 \( 1 - 1.23T + 13T^{2} \)
17 \( 1 + 0.810T + 17T^{2} \)
19 \( 1 - 6.55T + 19T^{2} \)
23 \( 1 + 5.53T + 23T^{2} \)
29 \( 1 + 0.0447T + 29T^{2} \)
31 \( 1 - 7.20T + 31T^{2} \)
37 \( 1 - 5.22T + 37T^{2} \)
43 \( 1 + 8.64T + 43T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 + 5.99T + 53T^{2} \)
59 \( 1 + 4.34T + 59T^{2} \)
61 \( 1 - 1.13T + 61T^{2} \)
67 \( 1 + 2.70T + 67T^{2} \)
71 \( 1 + 5.56T + 71T^{2} \)
73 \( 1 + 14.5T + 73T^{2} \)
79 \( 1 - 15.5T + 79T^{2} \)
83 \( 1 - 9.02T + 83T^{2} \)
89 \( 1 - 7.38T + 89T^{2} \)
97 \( 1 - 5.44T + 97T^{2} \)
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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

−7.85199505949303421605554788739, −7.54597959886073528951687197056, −6.51719138437183906319772182567, −5.60462312094952095523264393046, −4.81793459562122241311099604428, −3.98878942321070535952147864138, −2.96520705918886601581184867912, −2.18859080783189454466413911988, −1.19611426158060910040759906020, 0, 1.19611426158060910040759906020, 2.18859080783189454466413911988, 2.96520705918886601581184867912, 3.98878942321070535952147864138, 4.81793459562122241311099604428, 5.60462312094952095523264393046, 6.51719138437183906319772182567, 7.54597959886073528951687197056, 7.85199505949303421605554788739

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