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  + 4·7-s − 3·9-s + 6·11-s − 2·13-s + 6·17-s − 6·19-s + 23-s − 5·25-s − 6·29-s − 8·37-s + 6·41-s − 2·43-s − 8·47-s + 9·49-s − 8·53-s + 4·59-s − 4·61-s − 12·63-s + 2·67-s − 8·71-s + 6·73-s + 24·77-s + 12·79-s + 9·81-s + 10·83-s + 10·89-s − 8·91-s + ⋯
L(s)  = 1  + 1.51·7-s − 9-s + 1.80·11-s − 0.554·13-s + 1.45·17-s − 1.37·19-s + 0.208·23-s − 25-s − 1.11·29-s − 1.31·37-s + 0.937·41-s − 0.304·43-s − 1.16·47-s + 9/7·49-s − 1.09·53-s + 0.520·59-s − 0.512·61-s − 1.51·63-s + 0.244·67-s − 0.949·71-s + 0.702·73-s + 2.73·77-s + 1.35·79-s + 81-s + 1.09·83-s + 1.05·89-s − 0.838·91-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  =  $(2,\ 184,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.298974796$
$L(\frac12)$  $\approx$  $1.298974796$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−19.59419690409467, −19.07445452000706, −17.67543425694155, −17.21637489452813, −16.61656852149865, −14.83815999633959, −14.65289186646229, −13.88931466277453, −12.30242903509009, −11.65285962273060, −10.90027717989387, −9.505020586152698, −8.532719042369231, −7.651530692901097, −6.232145762714616, −5.075272738449825, −3.760291214912559, −1.776321846225180, 1.776321846225180, 3.760291214912559, 5.075272738449825, 6.232145762714616, 7.651530692901097, 8.532719042369231, 9.505020586152698, 10.90027717989387, 11.65285962273060, 12.30242903509009, 13.88931466277453, 14.65289186646229, 14.83815999633959, 16.61656852149865, 17.21637489452813, 17.67543425694155, 19.07445452000706, 19.59419690409467

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