Properties

Degree 2
Conductor $ 3^{2} \cdot 11 \cdot 61 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.57·2-s + 4.60·4-s − 4.00·5-s + 2.04·7-s − 6.70·8-s + 10.3·10-s + 11-s − 4.75·13-s − 5.25·14-s + 8.02·16-s + 0.339·17-s + 4.54·19-s − 18.4·20-s − 2.57·22-s + 4.75·23-s + 11.0·25-s + 12.2·26-s + 9.41·28-s + 4.28·29-s + 1.81·31-s − 7.22·32-s − 0.873·34-s − 8.18·35-s − 2.11·37-s − 11.6·38-s + 26.8·40-s + 10.2·41-s + ⋯
L(s)  = 1  − 1.81·2-s + 2.30·4-s − 1.79·5-s + 0.772·7-s − 2.37·8-s + 3.25·10-s + 0.301·11-s − 1.31·13-s − 1.40·14-s + 2.00·16-s + 0.0823·17-s + 1.04·19-s − 4.13·20-s − 0.548·22-s + 0.992·23-s + 2.21·25-s + 2.39·26-s + 1.77·28-s + 0.795·29-s + 0.326·31-s − 1.27·32-s − 0.149·34-s − 1.38·35-s − 0.347·37-s − 1.89·38-s + 4.25·40-s + 1.60·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6039\)    =    \(3^{2} \cdot 11 \cdot 61\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6039} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6039,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5013975342$
$L(\frac12)$  $\approx$  $0.5013975342$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;11,\;61\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;11,\;61\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
11 \( 1 - T \)
61 \( 1 - T \)
good2 \( 1 + 2.57T + 2T^{2} \)
5 \( 1 + 4.00T + 5T^{2} \)
7 \( 1 - 2.04T + 7T^{2} \)
13 \( 1 + 4.75T + 13T^{2} \)
17 \( 1 - 0.339T + 17T^{2} \)
19 \( 1 - 4.54T + 19T^{2} \)
23 \( 1 - 4.75T + 23T^{2} \)
29 \( 1 - 4.28T + 29T^{2} \)
31 \( 1 - 1.81T + 31T^{2} \)
37 \( 1 + 2.11T + 37T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 + 9.89T + 43T^{2} \)
47 \( 1 - 5.22T + 47T^{2} \)
53 \( 1 + 9.68T + 53T^{2} \)
59 \( 1 - 2.04T + 59T^{2} \)
67 \( 1 - 5.30T + 67T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 + 4.51T + 73T^{2} \)
79 \( 1 - 5.49T + 79T^{2} \)
83 \( 1 + 14.2T + 83T^{2} \)
89 \( 1 - 6.05T + 89T^{2} \)
97 \( 1 - 13.6T + 97T^{2} \)
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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

−8.056407184999887840722771260695, −7.49296764850059630123461037322, −7.26020488557451619408747275615, −6.44063279974050289468422421334, −5.09098974374675222306712795870, −4.49456599278551813702353734650, −3.32373318805088447573498242187, −2.62655868868884355334759639565, −1.37191044461190353140393294081, −0.53416547355162579861083366993, 0.53416547355162579861083366993, 1.37191044461190353140393294081, 2.62655868868884355334759639565, 3.32373318805088447573498242187, 4.49456599278551813702353734650, 5.09098974374675222306712795870, 6.44063279974050289468422421334, 7.26020488557451619408747275615, 7.49296764850059630123461037322, 8.056407184999887840722771260695

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