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 − 4.35·5-s + 3.05·7-s + 8-s − 4.35·10-s + 0.734·11-s − 2.43·13-s + 3.05·14-s + 16-s + 0.681·17-s − 1.87·19-s − 4.35·20-s + 0.734·22-s − 6.91·23-s + 13.9·25-s − 2.43·26-s + 3.05·28-s − 0.113·29-s + 1.56·31-s + 32-s + 0.681·34-s − 13.3·35-s − 3.71·37-s − 1.87·38-s − 4.35·40-s + 11.1·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.94·5-s + 1.15·7-s + 0.353·8-s − 1.37·10-s + 0.221·11-s − 0.675·13-s + 0.816·14-s + 0.250·16-s + 0.165·17-s − 0.430·19-s − 0.973·20-s + 0.156·22-s − 1.44·23-s + 2.79·25-s − 0.477·26-s + 0.577·28-s − 0.0210·29-s + 0.281·31-s + 0.176·32-s + 0.116·34-s − 2.25·35-s − 0.611·37-s − 0.304·38-s − 0.688·40-s + 1.73·41-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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
223 \( 1 + T \)
good5 \( 1 + 4.35T + 5T^{2} \)
7 \( 1 - 3.05T + 7T^{2} \)
11 \( 1 - 0.734T + 11T^{2} \)
13 \( 1 + 2.43T + 13T^{2} \)
17 \( 1 - 0.681T + 17T^{2} \)
19 \( 1 + 1.87T + 19T^{2} \)
23 \( 1 + 6.91T + 23T^{2} \)
29 \( 1 + 0.113T + 29T^{2} \)
31 \( 1 - 1.56T + 31T^{2} \)
37 \( 1 + 3.71T + 37T^{2} \)
41 \( 1 - 11.1T + 41T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 - 6.22T + 47T^{2} \)
53 \( 1 + 6.47T + 53T^{2} \)
59 \( 1 + 7.94T + 59T^{2} \)
61 \( 1 + 4.54T + 61T^{2} \)
67 \( 1 + 2.23T + 67T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 + 3.30T + 73T^{2} \)
79 \( 1 + 3.98T + 79T^{2} \)
83 \( 1 - 9.77T + 83T^{2} \)
89 \( 1 - 3.23T + 89T^{2} \)
97 \( 1 + 11.1T + 97T^{2} \)
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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

−8.006023430992368384880759557122, −7.45535136165714044705727743445, −6.74432647671267327720480468040, −5.70566330189309827559873563861, −4.69249384697693374410123973765, −4.39025823189463432230473701386, −3.65366665827196988532709149497, −2.70708682476567549062205826560, −1.49385049171834636435184798567, 0, 1.49385049171834636435184798567, 2.70708682476567549062205826560, 3.65366665827196988532709149497, 4.39025823189463432230473701386, 4.69249384697693374410123973765, 5.70566330189309827559873563861, 6.74432647671267327720480468040, 7.45535136165714044705727743445, 8.006023430992368384880759557122

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