Properties

Degree 2
Conductor $ 2^{3} \cdot 23 $
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 − 4·7-s − 2·9-s − 2·11-s + 7·13-s + 2·15-s − 4·17-s − 6·19-s + 4·21-s − 23-s − 25-s + 5·27-s + 5·29-s + 3·31-s + 2·33-s + 8·35-s + 2·37-s − 7·39-s − 9·41-s + 8·43-s + 4·45-s − 47-s + 9·49-s + 4·51-s − 6·53-s + 4·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s − 1.51·7-s − 2/3·9-s − 0.603·11-s + 1.94·13-s + 0.516·15-s − 0.970·17-s − 1.37·19-s + 0.872·21-s − 0.208·23-s − 1/5·25-s + 0.962·27-s + 0.928·29-s + 0.538·31-s + 0.348·33-s + 1.35·35-s + 0.328·37-s − 1.12·39-s − 1.40·41-s + 1.21·43-s + 0.596·45-s − 0.145·47-s + 9/7·49-s + 0.560·51-s − 0.824·53-s + 0.539·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(184\)    =    \(2^{3} \cdot 23\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{184} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 184,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;23\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;23\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
23 \( 1 + T \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−19.95482288906215, −19.31109151779073, −18.52390199427880, −17.54367542488708, −16.50824351620304, −15.82220665484238, −15.37698110112896, −13.75927658216061, −13.03076914855899, −12.10595891986664, −11.08905176524511, −10.46142285712534, −8.957303289299283, −8.175491441545991, −6.546393640427463, −6.048556103659449, −4.273320972502453, −3.077917203089919, 0, 3.077917203089919, 4.273320972502453, 6.048556103659449, 6.546393640427463, 8.175491441545991, 8.957303289299283, 10.46142285712534, 11.08905176524511, 12.10595891986664, 13.03076914855899, 13.75927658216061, 15.37698110112896, 15.82220665484238, 16.50824351620304, 17.54367542488708, 18.52390199427880, 19.31109151779073, 19.95482288906215

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