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 − 3.82·5-s + 3.22·7-s + 9-s + 3.21·11-s − 3.97·13-s − 3.82·15-s − 2.27·17-s + 0.0375·19-s + 3.22·21-s + 1.29·23-s + 9.66·25-s + 27-s − 4.68·29-s − 4.65·31-s + 3.21·33-s − 12.3·35-s − 2.10·37-s − 3.97·39-s − 9.37·41-s + 10.1·43-s − 3.82·45-s − 9.03·47-s + 3.37·49-s − 2.27·51-s − 0.602·53-s − 12.3·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.71·5-s + 1.21·7-s + 0.333·9-s + 0.970·11-s − 1.10·13-s − 0.988·15-s − 0.552·17-s + 0.00860·19-s + 0.702·21-s + 0.269·23-s + 1.93·25-s + 0.192·27-s − 0.870·29-s − 0.835·31-s + 0.560·33-s − 2.08·35-s − 0.346·37-s − 0.636·39-s − 1.46·41-s + 1.55·43-s − 0.570·45-s − 1.31·47-s + 0.481·49-s − 0.319·51-s − 0.0828·53-s − 1.66·55-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 \( 1 + 3.82T + 5T^{2} \)
7 \( 1 - 3.22T + 7T^{2} \)
11 \( 1 - 3.21T + 11T^{2} \)
13 \( 1 + 3.97T + 13T^{2} \)
17 \( 1 + 2.27T + 17T^{2} \)
19 \( 1 - 0.0375T + 19T^{2} \)
23 \( 1 - 1.29T + 23T^{2} \)
29 \( 1 + 4.68T + 29T^{2} \)
31 \( 1 + 4.65T + 31T^{2} \)
37 \( 1 + 2.10T + 37T^{2} \)
41 \( 1 + 9.37T + 41T^{2} \)
43 \( 1 - 10.1T + 43T^{2} \)
47 \( 1 + 9.03T + 47T^{2} \)
53 \( 1 + 0.602T + 53T^{2} \)
59 \( 1 - 3.58T + 59T^{2} \)
61 \( 1 + 3.24T + 61T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 + 5.99T + 71T^{2} \)
73 \( 1 - 0.896T + 73T^{2} \)
79 \( 1 - 2.02T + 79T^{2} \)
83 \( 1 - 0.335T + 83T^{2} \)
89 \( 1 - 6.96T + 89T^{2} \)
97 \( 1 + 5.63T + 97T^{2} \)
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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

−7.75834344945657938596452010568, −7.23750077875944219479252611582, −6.65660218984571467075188880912, −5.25667550290995654853918898530, −4.67262333237170177870799542033, −4.00378844430792608323824922902, −3.44260089303418382175266424552, −2.32724918308872616838923837090, −1.36566509646770508664526433346, 0, 1.36566509646770508664526433346, 2.32724918308872616838923837090, 3.44260089303418382175266424552, 4.00378844430792608323824922902, 4.67262333237170177870799542033, 5.25667550290995654853918898530, 6.65660218984571467075188880912, 7.23750077875944219479252611582, 7.75834344945657938596452010568

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