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 − 2.60·5-s − 1.53·7-s + 9-s + 4.96·11-s + 3.57·13-s + 2.60·15-s − 5.67·17-s + 7.50·19-s + 1.53·21-s − 5.35·23-s + 1.78·25-s − 27-s + 3.66·29-s + 4.23·31-s − 4.96·33-s + 3.98·35-s − 6.07·37-s − 3.57·39-s − 1.00·41-s + 1.75·43-s − 2.60·45-s − 2.20·47-s − 4.65·49-s + 5.67·51-s + 5.12·53-s − 12.9·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.16·5-s − 0.578·7-s + 0.333·9-s + 1.49·11-s + 0.991·13-s + 0.672·15-s − 1.37·17-s + 1.72·19-s + 0.334·21-s − 1.11·23-s + 0.356·25-s − 0.192·27-s + 0.681·29-s + 0.759·31-s − 0.865·33-s + 0.674·35-s − 0.998·37-s − 0.572·39-s − 0.156·41-s + 0.267·43-s − 0.388·45-s − 0.322·47-s − 0.665·49-s + 0.794·51-s + 0.703·53-s − 1.74·55-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  =  $(2,\ 6036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.123023060$
$L(\frac12)$  $\approx$  $1.123023060$
$L(\frac{3}{2})$   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 \( 1 + 2.60T + 5T^{2} \)
7 \( 1 + 1.53T + 7T^{2} \)
11 \( 1 - 4.96T + 11T^{2} \)
13 \( 1 - 3.57T + 13T^{2} \)
17 \( 1 + 5.67T + 17T^{2} \)
19 \( 1 - 7.50T + 19T^{2} \)
23 \( 1 + 5.35T + 23T^{2} \)
29 \( 1 - 3.66T + 29T^{2} \)
31 \( 1 - 4.23T + 31T^{2} \)
37 \( 1 + 6.07T + 37T^{2} \)
41 \( 1 + 1.00T + 41T^{2} \)
43 \( 1 - 1.75T + 43T^{2} \)
47 \( 1 + 2.20T + 47T^{2} \)
53 \( 1 - 5.12T + 53T^{2} \)
59 \( 1 - 1.75T + 59T^{2} \)
61 \( 1 - 2.51T + 61T^{2} \)
67 \( 1 - 12.8T + 67T^{2} \)
71 \( 1 + 7.52T + 71T^{2} \)
73 \( 1 + 0.588T + 73T^{2} \)
79 \( 1 - 3.18T + 79T^{2} \)
83 \( 1 + 1.21T + 83T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 + 16.6T + 97T^{2} \)
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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

−8.139030623217003442761835847243, −7.17600517053596195124399260580, −6.66404265094016103337094321151, −6.13859277493726223242682604130, −5.19537901933541224503614574792, −4.15489448673489971443650555537, −3.89361751899870652599553786417, −3.03392205682969235217423767354, −1.58987345050958878255484150900, −0.60077304548996441154510847183, 0.60077304548996441154510847183, 1.58987345050958878255484150900, 3.03392205682969235217423767354, 3.89361751899870652599553786417, 4.15489448673489971443650555537, 5.19537901933541224503614574792, 6.13859277493726223242682604130, 6.66404265094016103337094321151, 7.17600517053596195124399260580, 8.139030623217003442761835847243

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