Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 7 \cdot 11 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s + 7-s + 9-s + 11-s − 6·13-s − 2·15-s − 2·17-s − 8·19-s − 21-s − 4·23-s − 25-s − 27-s + 2·29-s − 8·31-s − 33-s + 2·35-s + 6·37-s + 6·39-s − 2·41-s + 8·43-s + 2·45-s − 4·47-s + 49-s + 2·51-s + 2·53-s + 2·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.66·13-s − 0.516·15-s − 0.485·17-s − 1.83·19-s − 0.218·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.174·33-s + 0.338·35-s + 0.986·37-s + 0.960·39-s − 0.312·41-s + 1.21·43-s + 0.298·45-s − 0.583·47-s + 1/7·49-s + 0.280·51-s + 0.274·53-s + 0.269·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1848\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1848} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1848,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−19.76877828783730, −19.30903586552156, −18.42946164689326, −17.61019613753856, −17.47048263481832, −16.75646117649545, −16.16993015614542, −14.99089879200262, −14.75202285961887, −13.93237788812982, −13.11705883084058, −12.49687179302666, −11.88849408469277, −10.97373072489699, −10.40258982766878, −9.656379210170952, −9.044257212083537, −8.004405194882106, −7.177502511865883, −6.333376003690149, −5.714706168972305, −4.770496136327717, −4.118892992162284, −2.464170177769671, −1.804761335648700, 0, 1.804761335648700, 2.464170177769671, 4.118892992162284, 4.770496136327717, 5.714706168972305, 6.333376003690149, 7.177502511865883, 8.004405194882106, 9.044257212083537, 9.656379210170952, 10.40258982766878, 10.97373072489699, 11.88849408469277, 12.49687179302666, 13.11705883084058, 13.93237788812982, 14.75202285961887, 14.99089879200262, 16.16993015614542, 16.75646117649545, 17.47048263481832, 17.61019613753856, 18.42946164689326, 19.30903586552156, 19.76877828783730

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