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 + 2.52·5-s − 1.02·7-s + 9-s − 3.26·11-s − 2.87·13-s + 2.52·15-s + 0.377·17-s − 1.62·19-s − 1.02·21-s − 6.17·23-s + 1.35·25-s + 27-s − 5.77·29-s + 6.59·31-s − 3.26·33-s − 2.58·35-s + 2.04·37-s − 2.87·39-s + 1.64·41-s − 1.25·43-s + 2.52·45-s + 12.2·47-s − 5.94·49-s + 0.377·51-s − 8.59·53-s − 8.23·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.12·5-s − 0.387·7-s + 0.333·9-s − 0.985·11-s − 0.797·13-s + 0.650·15-s + 0.0916·17-s − 0.371·19-s − 0.223·21-s − 1.28·23-s + 0.271·25-s + 0.192·27-s − 1.07·29-s + 1.18·31-s − 0.568·33-s − 0.436·35-s + 0.336·37-s − 0.460·39-s + 0.256·41-s − 0.191·43-s + 0.375·45-s + 1.78·47-s − 0.849·49-s + 0.0529·51-s − 1.18·53-s − 1.11·55-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 \( 1 - 2.52T + 5T^{2} \)
7 \( 1 + 1.02T + 7T^{2} \)
11 \( 1 + 3.26T + 11T^{2} \)
13 \( 1 + 2.87T + 13T^{2} \)
17 \( 1 - 0.377T + 17T^{2} \)
19 \( 1 + 1.62T + 19T^{2} \)
23 \( 1 + 6.17T + 23T^{2} \)
29 \( 1 + 5.77T + 29T^{2} \)
31 \( 1 - 6.59T + 31T^{2} \)
37 \( 1 - 2.04T + 37T^{2} \)
41 \( 1 - 1.64T + 41T^{2} \)
43 \( 1 + 1.25T + 43T^{2} \)
47 \( 1 - 12.2T + 47T^{2} \)
53 \( 1 + 8.59T + 53T^{2} \)
59 \( 1 + 9.35T + 59T^{2} \)
61 \( 1 + 1.13T + 61T^{2} \)
67 \( 1 - 0.942T + 67T^{2} \)
71 \( 1 - 0.988T + 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 + 3.81T + 79T^{2} \)
83 \( 1 - 4.87T + 83T^{2} \)
89 \( 1 + 16.9T + 89T^{2} \)
97 \( 1 - 5.17T + 97T^{2} \)
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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

−7.79942284282331077097150221070, −7.08975015840372897118281036993, −6.15433868961819158960876428401, −5.72969969477377825198717997387, −4.83904369955091449851662778380, −4.05646712508937365978280115448, −2.93490305949651019045511405024, −2.40542409684536935116215416149, −1.61592603011217542881398785583, 0, 1.61592603011217542881398785583, 2.40542409684536935116215416149, 2.93490305949651019045511405024, 4.05646712508937365978280115448, 4.83904369955091449851662778380, 5.72969969477377825198717997387, 6.15433868961819158960876428401, 7.08975015840372897118281036993, 7.79942284282331077097150221070

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