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 − 4.35·5-s + 4.52·7-s + 9-s − 1.05·11-s + 3.27·13-s + 4.35·15-s − 0.821·17-s + 6.20·19-s − 4.52·21-s + 4.58·23-s + 13.9·25-s − 27-s − 2.39·29-s + 3.22·31-s + 1.05·33-s − 19.7·35-s − 7.35·37-s − 3.27·39-s − 4.28·41-s + 4.67·43-s − 4.35·45-s + 9.26·47-s + 13.5·49-s + 0.821·51-s − 2.53·53-s + 4.60·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.94·5-s + 1.71·7-s + 0.333·9-s − 0.318·11-s + 0.907·13-s + 1.12·15-s − 0.199·17-s + 1.42·19-s − 0.988·21-s + 0.955·23-s + 2.78·25-s − 0.192·27-s − 0.444·29-s + 0.579·31-s + 0.184·33-s − 3.33·35-s − 1.20·37-s − 0.524·39-s − 0.668·41-s + 0.712·43-s − 0.648·45-s + 1.35·47-s + 1.93·49-s + 0.115·51-s − 0.348·53-s + 0.620·55-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  =  $(2,\ 6036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.426395955$
$L(\frac12)$  $\approx$  $1.426395955$
$L(\frac{3}{2})$   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 \( 1 + 4.35T + 5T^{2} \)
7 \( 1 - 4.52T + 7T^{2} \)
11 \( 1 + 1.05T + 11T^{2} \)
13 \( 1 - 3.27T + 13T^{2} \)
17 \( 1 + 0.821T + 17T^{2} \)
19 \( 1 - 6.20T + 19T^{2} \)
23 \( 1 - 4.58T + 23T^{2} \)
29 \( 1 + 2.39T + 29T^{2} \)
31 \( 1 - 3.22T + 31T^{2} \)
37 \( 1 + 7.35T + 37T^{2} \)
41 \( 1 + 4.28T + 41T^{2} \)
43 \( 1 - 4.67T + 43T^{2} \)
47 \( 1 - 9.26T + 47T^{2} \)
53 \( 1 + 2.53T + 53T^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 + 5.03T + 61T^{2} \)
67 \( 1 + 3.07T + 67T^{2} \)
71 \( 1 - 4.37T + 71T^{2} \)
73 \( 1 + 8.07T + 73T^{2} \)
79 \( 1 + 2.23T + 79T^{2} \)
83 \( 1 - 3.91T + 83T^{2} \)
89 \( 1 + 0.496T + 89T^{2} \)
97 \( 1 + 2.38T + 97T^{2} \)
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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

−7.938876857834110756063814207193, −7.46925541547925876128187697862, −6.97362002296240175960273075318, −5.77431207101634017237614433971, −4.97266335128718556721045339313, −4.58144349771045285271460486497, −3.77583197236766888195629024980, −3.01475759374746338666726842849, −1.50681063660028575822301822260, −0.71097408928670303933176293006, 0.71097408928670303933176293006, 1.50681063660028575822301822260, 3.01475759374746338666726842849, 3.77583197236766888195629024980, 4.58144349771045285271460486497, 4.97266335128718556721045339313, 5.77431207101634017237614433971, 6.97362002296240175960273075318, 7.46925541547925876128187697862, 7.938876857834110756063814207193

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