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 − 4·5-s + 2·7-s − 2·9-s − 4·11-s − 5·13-s + 4·15-s − 2·17-s + 6·19-s − 2·21-s + 23-s + 11·25-s + 5·27-s + 29-s − 9·31-s + 4·33-s − 8·35-s − 4·37-s + 5·39-s + 3·41-s + 8·43-s + 8·45-s − 5·47-s − 3·49-s + 2·51-s + 6·53-s + 16·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s + 0.755·7-s − 2/3·9-s − 1.20·11-s − 1.38·13-s + 1.03·15-s − 0.485·17-s + 1.37·19-s − 0.436·21-s + 0.208·23-s + 11/5·25-s + 0.962·27-s + 0.185·29-s − 1.61·31-s + 0.696·33-s − 1.35·35-s − 0.657·37-s + 0.800·39-s + 0.468·41-s + 1.21·43-s + 1.19·45-s − 0.729·47-s − 3/7·49-s + 0.280·51-s + 0.824·53-s + 2.15·55-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 \( 1 + 4 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 5 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 + 15 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−19.42426552205555, −18.33433735381099, −17.65775467076202, −16.55403273154620, −15.87649856361070, −15.01877496565173, −14.25823387068545, −12.74448244727547, −11.91079007041406, −11.32872114110074, −10.53801091148756, −8.910132789738256, −7.746819907992762, −7.336835336003804, −5.387340016993772, −4.607748496307536, −2.991221195778288, 0, 2.991221195778288, 4.607748496307536, 5.387340016993772, 7.336835336003804, 7.746819907992762, 8.910132789738256, 10.53801091148756, 11.32872114110074, 11.91079007041406, 12.74448244727547, 14.25823387068545, 15.01877496565173, 15.87649856361070, 16.55403273154620, 17.65775467076202, 18.33433735381099, 19.42426552205555

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