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 − 0.970·5-s + 3.74·7-s + 9-s − 5.32·11-s − 0.298·13-s + 0.970·15-s − 7.82·17-s − 3.73·19-s − 3.74·21-s + 1.49·23-s − 4.05·25-s − 27-s + 3.78·29-s − 2.76·31-s + 5.32·33-s − 3.63·35-s + 0.959·37-s + 0.298·39-s + 1.24·41-s + 0.457·43-s − 0.970·45-s + 8.90·47-s + 7.00·49-s + 7.82·51-s + 6.31·53-s + 5.16·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.433·5-s + 1.41·7-s + 0.333·9-s − 1.60·11-s − 0.0826·13-s + 0.250·15-s − 1.89·17-s − 0.856·19-s − 0.816·21-s + 0.312·23-s − 0.811·25-s − 0.192·27-s + 0.703·29-s − 0.496·31-s + 0.926·33-s − 0.613·35-s + 0.157·37-s + 0.0477·39-s + 0.194·41-s + 0.0697·43-s − 0.144·45-s + 1.29·47-s + 1.00·49-s + 1.09·51-s + 0.866·53-s + 0.696·55-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  =  $(2,\ 6036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.040402653$
$L(\frac12)$  $\approx$  $1.040402653$
$L(\frac{3}{2})$   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 \( 1 + 0.970T + 5T^{2} \)
7 \( 1 - 3.74T + 7T^{2} \)
11 \( 1 + 5.32T + 11T^{2} \)
13 \( 1 + 0.298T + 13T^{2} \)
17 \( 1 + 7.82T + 17T^{2} \)
19 \( 1 + 3.73T + 19T^{2} \)
23 \( 1 - 1.49T + 23T^{2} \)
29 \( 1 - 3.78T + 29T^{2} \)
31 \( 1 + 2.76T + 31T^{2} \)
37 \( 1 - 0.959T + 37T^{2} \)
41 \( 1 - 1.24T + 41T^{2} \)
43 \( 1 - 0.457T + 43T^{2} \)
47 \( 1 - 8.90T + 47T^{2} \)
53 \( 1 - 6.31T + 53T^{2} \)
59 \( 1 + 3.00T + 59T^{2} \)
61 \( 1 + 2.73T + 61T^{2} \)
67 \( 1 - 14.8T + 67T^{2} \)
71 \( 1 + 6.26T + 71T^{2} \)
73 \( 1 - 7.46T + 73T^{2} \)
79 \( 1 - 5.85T + 79T^{2} \)
83 \( 1 - 4.58T + 83T^{2} \)
89 \( 1 - 9.45T + 89T^{2} \)
97 \( 1 - 17.7T + 97T^{2} \)
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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

−8.050131964283549947398977059932, −7.45991894188133027167821341048, −6.71710946107785266860420207115, −5.83366656362607434737140445210, −5.06021930229586706428765134145, −4.60912184194595469207869773170, −3.92664705218733966129741148812, −2.49078308492272640073167636855, −1.98238620492081475303571785715, −0.53047566190904852154934483446, 0.53047566190904852154934483446, 1.98238620492081475303571785715, 2.49078308492272640073167636855, 3.92664705218733966129741148812, 4.60912184194595469207869773170, 5.06021930229586706428765134145, 5.83366656362607434737140445210, 6.71710946107785266860420207115, 7.45991894188133027167821341048, 8.050131964283549947398977059932

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