Properties

Degree 2
Conductor $ 2 \cdot 7^{2} \cdot 41 $
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 + 2·3-s + 4-s − 2·5-s + 2·6-s + 8-s + 9-s − 2·10-s − 4·11-s + 2·12-s + 13-s − 4·15-s + 16-s + 18-s − 6·19-s − 2·20-s − 4·22-s − 2·23-s + 2·24-s − 25-s + 26-s − 4·27-s − 5·29-s − 4·30-s + 4·31-s + 32-s − 8·33-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s + 0.577·12-s + 0.277·13-s − 1.03·15-s + 1/4·16-s + 0.235·18-s − 1.37·19-s − 0.447·20-s − 0.852·22-s − 0.417·23-s + 0.408·24-s − 1/5·25-s + 0.196·26-s − 0.769·27-s − 0.928·29-s − 0.730·30-s + 0.718·31-s + 0.176·32-s − 1.39·33-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4018\)    =    \(2 \cdot 7^{2} \cdot 41\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4018} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4018,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;41\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;41\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
7 \( 1 \)
41 \( 1 - T \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−8.192032851112216973830186051376, −7.54054361859304036607750755648, −6.66951474604403914259980265722, −5.78657006035942566046847608494, −4.89236832560684339986415106799, −4.04228238736635998473333209074, −3.48843370538725787568465334940, −2.64600065094863566637987939235, −1.92117085377281792003959313709, 0, 1.92117085377281792003959313709, 2.64600065094863566637987939235, 3.48843370538725787568465334940, 4.04228238736635998473333209074, 4.89236832560684339986415106799, 5.78657006035942566046847608494, 6.66951474604403914259980265722, 7.54054361859304036607750755648, 8.192032851112216973830186051376

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