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 + 1.90·5-s − 3.61·7-s + 9-s + 5.57·11-s + 5.61·13-s − 1.90·15-s + 6.17·17-s − 0.419·19-s + 3.61·21-s + 3.21·23-s − 1.37·25-s − 27-s + 3.62·29-s + 5.39·31-s − 5.57·33-s − 6.88·35-s − 2.73·37-s − 5.61·39-s − 2.39·41-s + 2.76·43-s + 1.90·45-s + 0.129·47-s + 6.09·49-s − 6.17·51-s + 8.82·53-s + 10.6·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.851·5-s − 1.36·7-s + 0.333·9-s + 1.68·11-s + 1.55·13-s − 0.491·15-s + 1.49·17-s − 0.0961·19-s + 0.789·21-s + 0.670·23-s − 0.275·25-s − 0.192·27-s + 0.673·29-s + 0.968·31-s − 0.970·33-s − 1.16·35-s − 0.449·37-s − 0.899·39-s − 0.373·41-s + 0.421·43-s + 0.283·45-s + 0.0188·47-s + 0.871·49-s − 0.864·51-s + 1.21·53-s + 1.43·55-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  =  $(2,\ 6036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.245930325$
$L(\frac12)$  $\approx$  $2.245930325$
$L(\frac{3}{2})$   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 \( 1 - 1.90T + 5T^{2} \)
7 \( 1 + 3.61T + 7T^{2} \)
11 \( 1 - 5.57T + 11T^{2} \)
13 \( 1 - 5.61T + 13T^{2} \)
17 \( 1 - 6.17T + 17T^{2} \)
19 \( 1 + 0.419T + 19T^{2} \)
23 \( 1 - 3.21T + 23T^{2} \)
29 \( 1 - 3.62T + 29T^{2} \)
31 \( 1 - 5.39T + 31T^{2} \)
37 \( 1 + 2.73T + 37T^{2} \)
41 \( 1 + 2.39T + 41T^{2} \)
43 \( 1 - 2.76T + 43T^{2} \)
47 \( 1 - 0.129T + 47T^{2} \)
53 \( 1 - 8.82T + 53T^{2} \)
59 \( 1 + 0.0305T + 59T^{2} \)
61 \( 1 + 3.48T + 61T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 - 3.48T + 71T^{2} \)
73 \( 1 + 7.80T + 73T^{2} \)
79 \( 1 + 14.5T + 79T^{2} \)
83 \( 1 - 5.10T + 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + 11.7T + 97T^{2} \)
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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

−8.141889064389430901151162060667, −6.99660044970916908144430936892, −6.53024676848586301074658304149, −5.98707257223577008149715106250, −5.58380879687559635198248783196, −4.33579917956138740119302307197, −3.60656304306274836265319660672, −2.97369902137980433633331000960, −1.52909800837393569886928693935, −0.911381265721873903418492593212, 0.911381265721873903418492593212, 1.52909800837393569886928693935, 2.97369902137980433633331000960, 3.60656304306274836265319660672, 4.33579917956138740119302307197, 5.58380879687559635198248783196, 5.98707257223577008149715106250, 6.53024676848586301074658304149, 6.99660044970916908144430936892, 8.141889064389430901151162060667

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