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 − 0.882·5-s − 2.85·7-s + 9-s + 2.43·11-s + 1.92·13-s − 0.882·15-s − 2.82·17-s + 5.97·19-s − 2.85·21-s − 3.88·23-s − 4.22·25-s + 27-s − 6.73·29-s + 0.533·31-s + 2.43·33-s + 2.52·35-s + 3.20·37-s + 1.92·39-s − 2.13·41-s − 7.27·43-s − 0.882·45-s − 0.965·47-s + 1.16·49-s − 2.82·51-s + 11.3·53-s − 2.15·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.394·5-s − 1.07·7-s + 0.333·9-s + 0.734·11-s + 0.533·13-s − 0.227·15-s − 0.686·17-s + 1.37·19-s − 0.623·21-s − 0.809·23-s − 0.844·25-s + 0.192·27-s − 1.25·29-s + 0.0958·31-s + 0.424·33-s + 0.426·35-s + 0.526·37-s + 0.308·39-s − 0.334·41-s − 1.11·43-s − 0.131·45-s − 0.140·47-s + 0.165·49-s − 0.396·51-s + 1.55·53-s − 0.290·55-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 \( 1 + 0.882T + 5T^{2} \)
7 \( 1 + 2.85T + 7T^{2} \)
11 \( 1 - 2.43T + 11T^{2} \)
13 \( 1 - 1.92T + 13T^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 - 5.97T + 19T^{2} \)
23 \( 1 + 3.88T + 23T^{2} \)
29 \( 1 + 6.73T + 29T^{2} \)
31 \( 1 - 0.533T + 31T^{2} \)
37 \( 1 - 3.20T + 37T^{2} \)
41 \( 1 + 2.13T + 41T^{2} \)
43 \( 1 + 7.27T + 43T^{2} \)
47 \( 1 + 0.965T + 47T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 + 8.38T + 59T^{2} \)
61 \( 1 + 4.72T + 61T^{2} \)
67 \( 1 - 2.35T + 67T^{2} \)
71 \( 1 + 6.46T + 71T^{2} \)
73 \( 1 - 11.4T + 73T^{2} \)
79 \( 1 + 3.91T + 79T^{2} \)
83 \( 1 + 12.5T + 83T^{2} \)
89 \( 1 - 14.9T + 89T^{2} \)
97 \( 1 + 9.34T + 97T^{2} \)
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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

−7.67283003193799514976341309443, −7.11070067145819192630483983602, −6.32928611636437863308298245710, −5.76187164257758031104052865215, −4.66106230809695522496786345623, −3.64552032806354457901921621983, −3.55002711205485855977922294722, −2.40955826455452552664092931560, −1.37410577090458877680436467261, 0, 1.37410577090458877680436467261, 2.40955826455452552664092931560, 3.55002711205485855977922294722, 3.64552032806354457901921621983, 4.66106230809695522496786345623, 5.76187164257758031104052865215, 6.32928611636437863308298245710, 7.11070067145819192630483983602, 7.67283003193799514976341309443

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