Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 503 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3.83·5-s + 0.0390·7-s + 9-s − 3.02·11-s + 2.65·13-s + 3.83·15-s − 1.40·17-s − 6.22·19-s − 0.0390·21-s − 6.84·23-s + 9.72·25-s − 27-s − 6.49·29-s + 5.46·31-s + 3.02·33-s − 0.149·35-s + 5.51·37-s − 2.65·39-s + 1.57·41-s − 7.27·43-s − 3.83·45-s − 8.27·47-s − 6.99·49-s + 1.40·51-s + 1.95·53-s + 11.6·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.71·5-s + 0.0147·7-s + 0.333·9-s − 0.913·11-s + 0.736·13-s + 0.990·15-s − 0.340·17-s − 1.42·19-s − 0.00852·21-s − 1.42·23-s + 1.94·25-s − 0.192·27-s − 1.20·29-s + 0.981·31-s + 0.527·33-s − 0.0253·35-s + 0.905·37-s − 0.425·39-s + 0.245·41-s − 1.10·43-s − 0.572·45-s − 1.20·47-s − 0.999·49-s + 0.196·51-s + 0.268·53-s + 1.56·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6036\)    =    \(2^{2} \cdot 3 \cdot 503\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6036} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.2824617709$
$L(\frac12)$  $\approx$  $0.2824617709$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;503\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;503\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 + T \)
503 \( 1 + T \)
good5 \( 1 + 3.83T + 5T^{2} \)
7 \( 1 - 0.0390T + 7T^{2} \)
11 \( 1 + 3.02T + 11T^{2} \)
13 \( 1 - 2.65T + 13T^{2} \)
17 \( 1 + 1.40T + 17T^{2} \)
19 \( 1 + 6.22T + 19T^{2} \)
23 \( 1 + 6.84T + 23T^{2} \)
29 \( 1 + 6.49T + 29T^{2} \)
31 \( 1 - 5.46T + 31T^{2} \)
37 \( 1 - 5.51T + 37T^{2} \)
41 \( 1 - 1.57T + 41T^{2} \)
43 \( 1 + 7.27T + 43T^{2} \)
47 \( 1 + 8.27T + 47T^{2} \)
53 \( 1 - 1.95T + 53T^{2} \)
59 \( 1 + 7.20T + 59T^{2} \)
61 \( 1 + 11.8T + 61T^{2} \)
67 \( 1 + 4.49T + 67T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 - 14.1T + 79T^{2} \)
83 \( 1 + 2.15T + 83T^{2} \)
89 \( 1 - 14.6T + 89T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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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

−8.006110474344928313820943952822, −7.58782249485585178640567152634, −6.54071592212896318005750610526, −6.13673494895870839234590697048, −5.03192162151192482942508448817, −4.37012858963774299799906234823, −3.84585514095691629304876795936, −2.94830131846616449196152319429, −1.74369940897620378661323861592, −0.27944394733648782595891829883, 0.27944394733648782595891829883, 1.74369940897620378661323861592, 2.94830131846616449196152319429, 3.84585514095691629304876795936, 4.37012858963774299799906234823, 5.03192162151192482942508448817, 6.13673494895870839234590697048, 6.54071592212896318005750610526, 7.58782249485585178640567152634, 8.006110474344928313820943952822

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