Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 223 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 4.60·7-s + 8-s + 2.69·11-s + 1.69·13-s − 4.60·14-s + 16-s − 2.60·17-s + 4.30·19-s + 2.69·22-s − 4·23-s − 5·25-s + 1.69·26-s − 4.60·28-s + 5.30·29-s + 6.60·31-s + 32-s − 2.60·34-s + 4.60·37-s + 4.30·38-s − 4·41-s + 3.90·43-s + 2.69·44-s − 4·46-s + 0.697·47-s + 14.2·49-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.74·7-s + 0.353·8-s + 0.813·11-s + 0.470·13-s − 1.23·14-s + 0.250·16-s − 0.631·17-s + 0.987·19-s + 0.575·22-s − 0.834·23-s − 25-s + 0.332·26-s − 0.870·28-s + 0.984·29-s + 1.18·31-s + 0.176·32-s − 0.446·34-s + 0.757·37-s + 0.698·38-s − 0.624·41-s + 0.596·43-s + 0.406·44-s − 0.589·46-s + 0.101·47-s + 2.03·49-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4014\)    =    \(2 \cdot 3^{2} \cdot 223\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4014} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4014,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.535397100$
$L(\frac12)$  $\approx$  $2.535397100$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;223\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;223\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 \)
223 \( 1 + T \)
good5 \( 1 + 5T^{2} \)
7 \( 1 + 4.60T + 7T^{2} \)
11 \( 1 - 2.69T + 11T^{2} \)
13 \( 1 - 1.69T + 13T^{2} \)
17 \( 1 + 2.60T + 17T^{2} \)
19 \( 1 - 4.30T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 5.30T + 29T^{2} \)
31 \( 1 - 6.60T + 31T^{2} \)
37 \( 1 - 4.60T + 37T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 - 3.90T + 43T^{2} \)
47 \( 1 - 0.697T + 47T^{2} \)
53 \( 1 - 6.90T + 53T^{2} \)
59 \( 1 - 0.302T + 59T^{2} \)
61 \( 1 + 2.69T + 61T^{2} \)
67 \( 1 - 4.60T + 67T^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 + 8.90T + 73T^{2} \)
79 \( 1 - 8.51T + 79T^{2} \)
83 \( 1 + 5.21T + 83T^{2} \)
89 \( 1 - 17.2T + 89T^{2} \)
97 \( 1 + 0.788T + 97T^{2} \)
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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

−8.452512580828103690275563107163, −7.52128082438125781523221355795, −6.65292165646271728631785750404, −6.29405210868210541701229249654, −5.66554143417133303229295296261, −4.49819863725997226427627388022, −3.78655078297298587025780136077, −3.16256737062436172329747616053, −2.24763094074381032511929667268, −0.813117710041396841667166824850, 0.813117710041396841667166824850, 2.24763094074381032511929667268, 3.16256737062436172329747616053, 3.78655078297298587025780136077, 4.49819863725997226427627388022, 5.66554143417133303229295296261, 6.29405210868210541701229249654, 6.65292165646271728631785750404, 7.52128082438125781523221355795, 8.452512580828103690275563107163

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