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 + 0.344·5-s + 2.72·7-s + 9-s − 0.712·11-s + 6.42·13-s − 0.344·15-s − 5.96·17-s + 1.12·19-s − 2.72·21-s + 2.47·23-s − 4.88·25-s − 27-s + 1.35·29-s + 3.54·31-s + 0.712·33-s + 0.940·35-s + 8.77·37-s − 6.42·39-s + 3.02·41-s + 12.8·43-s + 0.344·45-s − 9.01·47-s + 0.450·49-s + 5.96·51-s − 5.26·53-s − 0.245·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.154·5-s + 1.03·7-s + 0.333·9-s − 0.214·11-s + 1.78·13-s − 0.0889·15-s − 1.44·17-s + 0.257·19-s − 0.595·21-s + 0.517·23-s − 0.976·25-s − 0.192·27-s + 0.252·29-s + 0.636·31-s + 0.124·33-s + 0.158·35-s + 1.44·37-s − 1.02·39-s + 0.472·41-s + 1.95·43-s + 0.0513·45-s − 1.31·47-s + 0.0642·49-s + 0.835·51-s − 0.723·53-s − 0.0331·55-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  =  $(2,\ 6036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.080270396$
$L(\frac12)$  $\approx$  $2.080270396$
$L(\frac{3}{2})$   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 \( 1 - 0.344T + 5T^{2} \)
7 \( 1 - 2.72T + 7T^{2} \)
11 \( 1 + 0.712T + 11T^{2} \)
13 \( 1 - 6.42T + 13T^{2} \)
17 \( 1 + 5.96T + 17T^{2} \)
19 \( 1 - 1.12T + 19T^{2} \)
23 \( 1 - 2.47T + 23T^{2} \)
29 \( 1 - 1.35T + 29T^{2} \)
31 \( 1 - 3.54T + 31T^{2} \)
37 \( 1 - 8.77T + 37T^{2} \)
41 \( 1 - 3.02T + 41T^{2} \)
43 \( 1 - 12.8T + 43T^{2} \)
47 \( 1 + 9.01T + 47T^{2} \)
53 \( 1 + 5.26T + 53T^{2} \)
59 \( 1 - 9.02T + 59T^{2} \)
61 \( 1 - 7.78T + 61T^{2} \)
67 \( 1 + 12.5T + 67T^{2} \)
71 \( 1 - 4.01T + 71T^{2} \)
73 \( 1 - 8.39T + 73T^{2} \)
79 \( 1 + 1.26T + 79T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 - 0.273T + 89T^{2} \)
97 \( 1 + 0.105T + 97T^{2} \)
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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

−8.092142697443849080566015549357, −7.42394459928526970590703059308, −6.41113018670872298591712217088, −6.07138076519907166166247601043, −5.20950263490056695915742054286, −4.45291605040224917296248314382, −3.89803802320985199495265695922, −2.66298586481151517461376066207, −1.69831068641184706248521263847, −0.828195383406123268083611027347, 0.828195383406123268083611027347, 1.69831068641184706248521263847, 2.66298586481151517461376066207, 3.89803802320985199495265695922, 4.45291605040224917296248314382, 5.20950263490056695915742054286, 6.07138076519907166166247601043, 6.41113018670872298591712217088, 7.42394459928526970590703059308, 8.092142697443849080566015549357

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