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 + 2.35·5-s − 1.22·7-s + 8-s + 2.35·10-s − 5.87·11-s − 3.29·13-s − 1.22·14-s + 16-s + 1.69·17-s − 0.613·19-s + 2.35·20-s − 5.87·22-s − 5.52·23-s + 0.523·25-s − 3.29·26-s − 1.22·28-s − 6.27·29-s − 1.38·31-s + 32-s + 1.69·34-s − 2.89·35-s + 2.93·37-s − 0.613·38-s + 2.35·40-s + 2.48·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.05·5-s − 0.464·7-s + 0.353·8-s + 0.743·10-s − 1.77·11-s − 0.913·13-s − 0.328·14-s + 0.250·16-s + 0.411·17-s − 0.140·19-s + 0.525·20-s − 1.25·22-s − 1.15·23-s + 0.104·25-s − 0.645·26-s − 0.232·28-s − 1.16·29-s − 0.248·31-s + 0.176·32-s + 0.290·34-s − 0.488·35-s + 0.481·37-s − 0.0995·38-s + 0.371·40-s + 0.388·41-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 \( 1 - 2.35T + 5T^{2} \)
7 \( 1 + 1.22T + 7T^{2} \)
11 \( 1 + 5.87T + 11T^{2} \)
13 \( 1 + 3.29T + 13T^{2} \)
17 \( 1 - 1.69T + 17T^{2} \)
19 \( 1 + 0.613T + 19T^{2} \)
23 \( 1 + 5.52T + 23T^{2} \)
29 \( 1 + 6.27T + 29T^{2} \)
31 \( 1 + 1.38T + 31T^{2} \)
37 \( 1 - 2.93T + 37T^{2} \)
41 \( 1 - 2.48T + 41T^{2} \)
43 \( 1 + 3.47T + 43T^{2} \)
47 \( 1 - 0.843T + 47T^{2} \)
53 \( 1 - 3.77T + 53T^{2} \)
59 \( 1 - 4.43T + 59T^{2} \)
61 \( 1 + 7.93T + 61T^{2} \)
67 \( 1 + 8.35T + 67T^{2} \)
71 \( 1 + 7.83T + 71T^{2} \)
73 \( 1 + 6.71T + 73T^{2} \)
79 \( 1 - 0.124T + 79T^{2} \)
83 \( 1 - 0.936T + 83T^{2} \)
89 \( 1 - 9.09T + 89T^{2} \)
97 \( 1 - 1.28T + 97T^{2} \)
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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

−7.74097875486068646881743005539, −7.46057003569035182606238991286, −6.32783990413348826683985525312, −5.73945024892319749640850179828, −5.24689427694212538791892563048, −4.40459894800188114180195460974, −3.28182365885607572341236543046, −2.50046153014486675826209355010, −1.86095785676571554135156501400, 0, 1.86095785676571554135156501400, 2.50046153014486675826209355010, 3.28182365885607572341236543046, 4.40459894800188114180195460974, 5.24689427694212538791892563048, 5.73945024892319749640850179828, 6.32783990413348826683985525312, 7.46057003569035182606238991286, 7.74097875486068646881743005539

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