Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 0.326·5-s − 1.59·7-s + 8-s − 0.326·10-s + 2.21·11-s + 2.49·13-s − 1.59·14-s + 16-s − 6.46·17-s − 7.66·19-s − 0.326·20-s + 2.21·22-s − 2.91·23-s − 4.89·25-s + 2.49·26-s − 1.59·28-s + 4.18·29-s + 5.66·31-s + 32-s − 6.46·34-s + 0.521·35-s − 2.05·37-s − 7.66·38-s − 0.326·40-s − 12.2·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.146·5-s − 0.603·7-s + 0.353·8-s − 0.103·10-s + 0.669·11-s + 0.690·13-s − 0.426·14-s + 0.250·16-s − 1.56·17-s − 1.75·19-s − 0.0730·20-s + 0.473·22-s − 0.607·23-s − 0.978·25-s + 0.488·26-s − 0.301·28-s + 0.777·29-s + 1.01·31-s + 0.176·32-s − 1.10·34-s + 0.0881·35-s − 0.337·37-s − 1.24·38-s − 0.0516·40-s − 1.91·41-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  =  1
Selberg data  =  $(2,\ 4014,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 0.326T + 5T^{2} \)
7 \( 1 + 1.59T + 7T^{2} \)
11 \( 1 - 2.21T + 11T^{2} \)
13 \( 1 - 2.49T + 13T^{2} \)
17 \( 1 + 6.46T + 17T^{2} \)
19 \( 1 + 7.66T + 19T^{2} \)
23 \( 1 + 2.91T + 23T^{2} \)
29 \( 1 - 4.18T + 29T^{2} \)
31 \( 1 - 5.66T + 31T^{2} \)
37 \( 1 + 2.05T + 37T^{2} \)
41 \( 1 + 12.2T + 41T^{2} \)
43 \( 1 + 8.89T + 43T^{2} \)
47 \( 1 - 8.26T + 47T^{2} \)
53 \( 1 - 6.20T + 53T^{2} \)
59 \( 1 - 0.283T + 59T^{2} \)
61 \( 1 + 2.94T + 61T^{2} \)
67 \( 1 + 5.67T + 67T^{2} \)
71 \( 1 + 4.12T + 71T^{2} \)
73 \( 1 - 7.69T + 73T^{2} \)
79 \( 1 + 0.116T + 79T^{2} \)
83 \( 1 - 7.08T + 83T^{2} \)
89 \( 1 + 0.513T + 89T^{2} \)
97 \( 1 + 6.38T + 97T^{2} \)
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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

−8.286874716225344955640888246328, −6.98606192686569470183794772314, −6.44905276105922374902736795130, −6.12619056656465431979251036947, −4.91432531589148299245896366837, −4.16206298161431499245167700811, −3.65585388575758733420682034485, −2.54419204484245494543720674953, −1.68481640021134302968970852400, 0, 1.68481640021134302968970852400, 2.54419204484245494543720674953, 3.65585388575758733420682034485, 4.16206298161431499245167700811, 4.91432531589148299245896366837, 6.12619056656465431979251036947, 6.44905276105922374902736795130, 6.98606192686569470183794772314, 8.286874716225344955640888246328

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