Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 503 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 0.754·5-s − 4.03·7-s + 9-s − 3.82·11-s − 4.25·13-s − 0.754·15-s − 5.34·17-s − 3.06·19-s + 4.03·21-s − 3.68·23-s − 4.43·25-s − 27-s − 6.36·29-s − 10.0·31-s + 3.82·33-s − 3.04·35-s + 9.28·37-s + 4.25·39-s + 2.62·41-s + 7.26·43-s + 0.754·45-s − 10.3·47-s + 9.26·49-s + 5.34·51-s + 10.2·53-s − 2.88·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.337·5-s − 1.52·7-s + 0.333·9-s − 1.15·11-s − 1.18·13-s − 0.194·15-s − 1.29·17-s − 0.704·19-s + 0.880·21-s − 0.768·23-s − 0.886·25-s − 0.192·27-s − 1.18·29-s − 1.81·31-s + 0.665·33-s − 0.514·35-s + 1.52·37-s + 0.681·39-s + 0.410·41-s + 1.10·43-s + 0.112·45-s − 1.51·47-s + 1.32·49-s + 0.748·51-s + 1.41·53-s − 0.388·55-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  =  0
Selberg data  =  $(2,\ 6036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.1384090128$
$L(\frac12)$  $\approx$  $0.1384090128$
$L(\frac{3}{2})$   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 \( 1 - 0.754T + 5T^{2} \)
7 \( 1 + 4.03T + 7T^{2} \)
11 \( 1 + 3.82T + 11T^{2} \)
13 \( 1 + 4.25T + 13T^{2} \)
17 \( 1 + 5.34T + 17T^{2} \)
19 \( 1 + 3.06T + 19T^{2} \)
23 \( 1 + 3.68T + 23T^{2} \)
29 \( 1 + 6.36T + 29T^{2} \)
31 \( 1 + 10.0T + 31T^{2} \)
37 \( 1 - 9.28T + 37T^{2} \)
41 \( 1 - 2.62T + 41T^{2} \)
43 \( 1 - 7.26T + 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 - 10.2T + 53T^{2} \)
59 \( 1 - 6.42T + 59T^{2} \)
61 \( 1 - 9.19T + 61T^{2} \)
67 \( 1 + 6.71T + 67T^{2} \)
71 \( 1 + 0.934T + 71T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 + 0.170T + 79T^{2} \)
83 \( 1 - 10.4T + 83T^{2} \)
89 \( 1 - 7.13T + 89T^{2} \)
97 \( 1 + 12.3T + 97T^{2} \)
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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

−7.87746826898543200205080734441, −7.26765656075648461113320381513, −6.62607887149198109910807715534, −5.85508906893820154363437362763, −5.46742731492026397377685324101, −4.41579524047146805170931929120, −3.74753260074943449756463508751, −2.55729800723850033258404445788, −2.11838261420643982407086584116, −0.18253035624982413178492872370, 0.18253035624982413178492872370, 2.11838261420643982407086584116, 2.55729800723850033258404445788, 3.74753260074943449756463508751, 4.41579524047146805170931929120, 5.46742731492026397377685324101, 5.85508906893820154363437362763, 6.62607887149198109910807715534, 7.26765656075648461113320381513, 7.87746826898543200205080734441

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