Properties

Degree 2
Conductor $ 2^{3} \cdot 23 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 2·7-s + 6·9-s − 5·13-s − 6·17-s + 6·19-s − 6·21-s + 23-s − 5·25-s + 9·27-s + 9·29-s + 3·31-s − 8·37-s − 15·39-s + 3·41-s − 8·43-s + 7·47-s − 3·49-s − 18·51-s − 2·53-s + 18·57-s + 4·59-s − 10·61-s − 12·63-s + 8·67-s + 3·69-s + 7·71-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.755·7-s + 2·9-s − 1.38·13-s − 1.45·17-s + 1.37·19-s − 1.30·21-s + 0.208·23-s − 25-s + 1.73·27-s + 1.67·29-s + 0.538·31-s − 1.31·37-s − 2.40·39-s + 0.468·41-s − 1.21·43-s + 1.02·47-s − 3/7·49-s − 2.52·51-s − 0.274·53-s + 2.38·57-s + 0.520·59-s − 1.28·61-s − 1.51·63-s + 0.977·67-s + 0.361·69-s + 0.830·71-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  =  0
Selberg data  =  $(2,\ 184,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.753106746$
$L(\frac12)$  $\approx$  $1.753106746$
$L(\frac{3}{2})$   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 \( 1 - p T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−19.72296982968294, −19.28481851616629, −18.18859432526581, −17.19535964531284, −15.63076972977079, −15.60934487429925, −14.25827738426122, −13.76553341591100, −12.87278196363648, −11.87035883342760, −10.16281162238177, −9.541607038853316, −8.669848375640950, −7.603645981972773, −6.714490483354076, −4.736299205549860, −3.352386031001376, −2.337089625113851, 2.337089625113851, 3.352386031001376, 4.736299205549860, 6.714490483354076, 7.603645981972773, 8.669848375640950, 9.541607038853316, 10.16281162238177, 11.87035883342760, 12.87278196363648, 13.76553341591100, 14.25827738426122, 15.60934487429925, 15.63076972977079, 17.19535964531284, 18.18859432526581, 19.28481851616629, 19.72296982968294

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