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 − 0.244·5-s + 2.69·7-s + 9-s + 4.17·11-s − 4.50·13-s + 0.244·15-s + 1.19·17-s + 2.55·19-s − 2.69·21-s + 4.82·23-s − 4.94·25-s − 27-s − 2.33·29-s + 8.83·31-s − 4.17·33-s − 0.659·35-s + 6.02·37-s + 4.50·39-s − 9.45·41-s + 7.71·43-s − 0.244·45-s + 2.80·47-s + 0.264·49-s − 1.19·51-s + 3.72·53-s − 1.02·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.109·5-s + 1.01·7-s + 0.333·9-s + 1.25·11-s − 1.24·13-s + 0.0632·15-s + 0.290·17-s + 0.586·19-s − 0.588·21-s + 1.00·23-s − 0.988·25-s − 0.192·27-s − 0.433·29-s + 1.58·31-s − 0.726·33-s − 0.111·35-s + 0.989·37-s + 0.721·39-s − 1.47·41-s + 1.17·43-s − 0.0364·45-s + 0.408·47-s + 0.0378·49-s − 0.167·51-s + 0.511·53-s − 0.137·55-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  =  $(2,\ 6036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.966071425$
$L(\frac12)$  $\approx$  $1.966071425$
$L(\frac{3}{2})$   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 \( 1 + 0.244T + 5T^{2} \)
7 \( 1 - 2.69T + 7T^{2} \)
11 \( 1 - 4.17T + 11T^{2} \)
13 \( 1 + 4.50T + 13T^{2} \)
17 \( 1 - 1.19T + 17T^{2} \)
19 \( 1 - 2.55T + 19T^{2} \)
23 \( 1 - 4.82T + 23T^{2} \)
29 \( 1 + 2.33T + 29T^{2} \)
31 \( 1 - 8.83T + 31T^{2} \)
37 \( 1 - 6.02T + 37T^{2} \)
41 \( 1 + 9.45T + 41T^{2} \)
43 \( 1 - 7.71T + 43T^{2} \)
47 \( 1 - 2.80T + 47T^{2} \)
53 \( 1 - 3.72T + 53T^{2} \)
59 \( 1 - 6.71T + 59T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 + 5.48T + 67T^{2} \)
71 \( 1 - 2.35T + 71T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 - 1.84T + 79T^{2} \)
83 \( 1 + 6.03T + 83T^{2} \)
89 \( 1 + 0.866T + 89T^{2} \)
97 \( 1 - 7.72T + 97T^{2} \)
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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

−7.906842607619735278195857654340, −7.39475798789342869257365699279, −6.69444079354327610490915259736, −5.90994096966200811241316137058, −5.10385982472117631644815726315, −4.58511115815435098012842036657, −3.83064463014156521437524700271, −2.71674446453759314120083286921, −1.67357902885938297346389867964, −0.806258213282553919356179206094, 0.806258213282553919356179206094, 1.67357902885938297346389867964, 2.71674446453759314120083286921, 3.83064463014156521437524700271, 4.58511115815435098012842036657, 5.10385982472117631644815726315, 5.90994096966200811241316137058, 6.69444079354327610490915259736, 7.39475798789342869257365699279, 7.906842607619735278195857654340

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