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 − 0.461·5-s − 3.07·7-s + 9-s − 5.13·11-s + 6.80·13-s + 0.461·15-s + 1.28·17-s + 6.12·19-s + 3.07·21-s − 8.36·23-s − 4.78·25-s − 27-s + 0.615·29-s − 3.68·31-s + 5.13·33-s + 1.41·35-s + 10.6·37-s − 6.80·39-s − 10.4·41-s − 0.274·43-s − 0.461·45-s + 7.41·47-s + 2.44·49-s − 1.28·51-s − 6.05·53-s + 2.36·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.206·5-s − 1.16·7-s + 0.333·9-s − 1.54·11-s + 1.88·13-s + 0.119·15-s + 0.311·17-s + 1.40·19-s + 0.670·21-s − 1.74·23-s − 0.957·25-s − 0.192·27-s + 0.114·29-s − 0.662·31-s + 0.893·33-s + 0.239·35-s + 1.75·37-s − 1.08·39-s − 1.62·41-s − 0.0418·43-s − 0.0687·45-s + 1.08·47-s + 0.349·49-s − 0.179·51-s − 0.832·53-s + 0.319·55-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  =  $(2,\ 6036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8730046545$
$L(\frac12)$  $\approx$  $0.8730046545$
$L(\frac{3}{2})$   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 \( 1 + 0.461T + 5T^{2} \)
7 \( 1 + 3.07T + 7T^{2} \)
11 \( 1 + 5.13T + 11T^{2} \)
13 \( 1 - 6.80T + 13T^{2} \)
17 \( 1 - 1.28T + 17T^{2} \)
19 \( 1 - 6.12T + 19T^{2} \)
23 \( 1 + 8.36T + 23T^{2} \)
29 \( 1 - 0.615T + 29T^{2} \)
31 \( 1 + 3.68T + 31T^{2} \)
37 \( 1 - 10.6T + 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 + 0.274T + 43T^{2} \)
47 \( 1 - 7.41T + 47T^{2} \)
53 \( 1 + 6.05T + 53T^{2} \)
59 \( 1 + 13.1T + 59T^{2} \)
61 \( 1 - 4.72T + 61T^{2} \)
67 \( 1 - 8.57T + 67T^{2} \)
71 \( 1 + 9.71T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 + 10.4T + 79T^{2} \)
83 \( 1 + 0.536T + 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 - 13.4T + 97T^{2} \)
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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

−7.85543013392053135492784990385, −7.54429735617179383518279709130, −6.43071907414653311471151133068, −5.90235975453580887289527282627, −5.51117556930505026001586070321, −4.38782325275494097148832262820, −3.55541130547346020732160986294, −3.01161381718475130375927578292, −1.74356393395924789103884120861, −0.49854455193298883356982225223, 0.49854455193298883356982225223, 1.74356393395924789103884120861, 3.01161381718475130375927578292, 3.55541130547346020732160986294, 4.38782325275494097148832262820, 5.51117556930505026001586070321, 5.90235975453580887289527282627, 6.43071907414653311471151133068, 7.54429735617179383518279709130, 7.85543013392053135492784990385

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