Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 1.92·5-s + 2.55·7-s + 9-s + 0.815·11-s + 5.13·13-s − 1.92·15-s − 6.73·17-s − 0.147·19-s + 2.55·21-s − 9.03·23-s − 1.28·25-s + 27-s − 7.65·29-s − 3.76·31-s + 0.815·33-s − 4.91·35-s − 4.31·37-s + 5.13·39-s + 10.2·41-s + 4.82·43-s − 1.92·45-s + 2.32·47-s − 0.495·49-s − 6.73·51-s − 10.7·53-s − 1.57·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.862·5-s + 0.963·7-s + 0.333·9-s + 0.245·11-s + 1.42·13-s − 0.497·15-s − 1.63·17-s − 0.0337·19-s + 0.556·21-s − 1.88·23-s − 0.256·25-s + 0.192·27-s − 1.42·29-s − 0.676·31-s + 0.141·33-s − 0.831·35-s − 0.709·37-s + 0.822·39-s + 1.60·41-s + 0.736·43-s − 0.287·45-s + 0.339·47-s − 0.0708·49-s − 0.943·51-s − 1.47·53-s − 0.211·55-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  =  1
Selberg data  =  $(2,\ 6036,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 1.92T + 5T^{2} \)
7 \( 1 - 2.55T + 7T^{2} \)
11 \( 1 - 0.815T + 11T^{2} \)
13 \( 1 - 5.13T + 13T^{2} \)
17 \( 1 + 6.73T + 17T^{2} \)
19 \( 1 + 0.147T + 19T^{2} \)
23 \( 1 + 9.03T + 23T^{2} \)
29 \( 1 + 7.65T + 29T^{2} \)
31 \( 1 + 3.76T + 31T^{2} \)
37 \( 1 + 4.31T + 37T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 - 4.82T + 43T^{2} \)
47 \( 1 - 2.32T + 47T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 + 4.54T + 59T^{2} \)
61 \( 1 - 2.14T + 61T^{2} \)
67 \( 1 + 8.28T + 67T^{2} \)
71 \( 1 + 5.89T + 71T^{2} \)
73 \( 1 + 3.28T + 73T^{2} \)
79 \( 1 - 5.54T + 79T^{2} \)
83 \( 1 + 9.06T + 83T^{2} \)
89 \( 1 + 0.924T + 89T^{2} \)
97 \( 1 - 14.2T + 97T^{2} \)
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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

−7.84064953647951377135375625460, −7.27126382727487289746298972742, −6.27776522470788758181093124325, −5.68143580885602930402730034508, −4.39902279095080768605602442845, −4.14558259671866002209604384257, −3.42740355089944424525871035054, −2.16448588238808329773340238288, −1.54178442154159973370982534611, 0, 1.54178442154159973370982534611, 2.16448588238808329773340238288, 3.42740355089944424525871035054, 4.14558259671866002209604384257, 4.39902279095080768605602442845, 5.68143580885602930402730034508, 6.27776522470788758181093124325, 7.27126382727487289746298972742, 7.84064953647951377135375625460

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