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 + 3.04·5-s + 4.50·7-s + 9-s + 3.45·11-s − 2.07·13-s − 3.04·15-s − 5.67·17-s + 7.79·19-s − 4.50·21-s − 9.14·23-s + 4.27·25-s − 27-s − 8.61·29-s + 0.690·31-s − 3.45·33-s + 13.7·35-s + 4.51·37-s + 2.07·39-s + 6.24·41-s + 10.9·43-s + 3.04·45-s + 6.37·47-s + 13.2·49-s + 5.67·51-s − 1.10·53-s + 10.5·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.36·5-s + 1.70·7-s + 0.333·9-s + 1.04·11-s − 0.576·13-s − 0.786·15-s − 1.37·17-s + 1.78·19-s − 0.982·21-s − 1.90·23-s + 0.854·25-s − 0.192·27-s − 1.60·29-s + 0.124·31-s − 0.602·33-s + 2.31·35-s + 0.741·37-s + 0.332·39-s + 0.975·41-s + 1.67·43-s + 0.453·45-s + 0.929·47-s + 1.89·49-s + 0.794·51-s − 0.151·53-s + 1.42·55-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  =  $(2,\ 6036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.888265212$
$L(\frac12)$  $\approx$  $2.888265212$
$L(\frac{3}{2})$   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 \( 1 - 3.04T + 5T^{2} \)
7 \( 1 - 4.50T + 7T^{2} \)
11 \( 1 - 3.45T + 11T^{2} \)
13 \( 1 + 2.07T + 13T^{2} \)
17 \( 1 + 5.67T + 17T^{2} \)
19 \( 1 - 7.79T + 19T^{2} \)
23 \( 1 + 9.14T + 23T^{2} \)
29 \( 1 + 8.61T + 29T^{2} \)
31 \( 1 - 0.690T + 31T^{2} \)
37 \( 1 - 4.51T + 37T^{2} \)
41 \( 1 - 6.24T + 41T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 - 6.37T + 47T^{2} \)
53 \( 1 + 1.10T + 53T^{2} \)
59 \( 1 + 12.6T + 59T^{2} \)
61 \( 1 - 8.83T + 61T^{2} \)
67 \( 1 - 6.27T + 67T^{2} \)
71 \( 1 - 2.91T + 71T^{2} \)
73 \( 1 + 4.14T + 73T^{2} \)
79 \( 1 + 2.05T + 79T^{2} \)
83 \( 1 - 16.8T + 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 + 2.33T + 97T^{2} \)
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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

−7.81467138901666975871263435210, −7.50849461657017572620342391951, −6.47867758875518544334973934457, −5.81936896764119077493148646551, −5.36310193442987712968638171365, −4.54047350602216640716061546603, −3.93928588247727914049513255876, −2.31335876380355760908562088039, −1.88716382241398095005925447997, −0.983002249500871371813378093838, 0.983002249500871371813378093838, 1.88716382241398095005925447997, 2.31335876380355760908562088039, 3.93928588247727914049513255876, 4.54047350602216640716061546603, 5.36310193442987712968638171365, 5.81936896764119077493148646551, 6.47867758875518544334973934457, 7.50849461657017572620342391951, 7.81467138901666975871263435210

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