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 + 0.336·5-s + 0.568·7-s + 9-s + 5.54·11-s − 5.92·13-s + 0.336·15-s + 0.0480·17-s − 3.03·19-s + 0.568·21-s − 5.32·23-s − 4.88·25-s + 27-s + 0.918·29-s − 8.65·31-s + 5.54·33-s + 0.191·35-s − 6.04·37-s − 5.92·39-s + 1.72·41-s − 6.69·43-s + 0.336·45-s − 0.136·47-s − 6.67·49-s + 0.0480·51-s − 1.99·53-s + 1.86·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.150·5-s + 0.214·7-s + 0.333·9-s + 1.67·11-s − 1.64·13-s + 0.0868·15-s + 0.0116·17-s − 0.695·19-s + 0.124·21-s − 1.11·23-s − 0.977·25-s + 0.192·27-s + 0.170·29-s − 1.55·31-s + 0.964·33-s + 0.0323·35-s − 0.994·37-s − 0.949·39-s + 0.269·41-s − 1.02·43-s + 0.0501·45-s − 0.0199·47-s − 0.953·49-s + 0.00673·51-s − 0.274·53-s + 0.251·55-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 \( 1 - 0.336T + 5T^{2} \)
7 \( 1 - 0.568T + 7T^{2} \)
11 \( 1 - 5.54T + 11T^{2} \)
13 \( 1 + 5.92T + 13T^{2} \)
17 \( 1 - 0.0480T + 17T^{2} \)
19 \( 1 + 3.03T + 19T^{2} \)
23 \( 1 + 5.32T + 23T^{2} \)
29 \( 1 - 0.918T + 29T^{2} \)
31 \( 1 + 8.65T + 31T^{2} \)
37 \( 1 + 6.04T + 37T^{2} \)
41 \( 1 - 1.72T + 41T^{2} \)
43 \( 1 + 6.69T + 43T^{2} \)
47 \( 1 + 0.136T + 47T^{2} \)
53 \( 1 + 1.99T + 53T^{2} \)
59 \( 1 + 11.7T + 59T^{2} \)
61 \( 1 - 6.84T + 61T^{2} \)
67 \( 1 - 7.26T + 67T^{2} \)
71 \( 1 - 12.1T + 71T^{2} \)
73 \( 1 + 5.77T + 73T^{2} \)
79 \( 1 + 15.6T + 79T^{2} \)
83 \( 1 + 5.52T + 83T^{2} \)
89 \( 1 + 7.27T + 89T^{2} \)
97 \( 1 - 6.99T + 97T^{2} \)
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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

−7.70796078262627282142326324936, −7.06329266475039528642798969779, −6.43932300936957773160564618894, −5.58792232085051383810809940292, −4.67247917598012338099795607594, −4.01898199272857782304571283321, −3.30487011334769625253441705395, −2.11401428398453786493827221554, −1.66971185223449705088234789797, 0, 1.66971185223449705088234789797, 2.11401428398453786493827221554, 3.30487011334769625253441705395, 4.01898199272857782304571283321, 4.67247917598012338099795607594, 5.58792232085051383810809940292, 6.43932300936957773160564618894, 7.06329266475039528642798969779, 7.70796078262627282142326324936

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