Properties

Degree 2
Conductor $ 2^{2} \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-s + 2·7-s − 2·9-s − 13-s − 6·17-s + 2·19-s + 2·21-s − 23-s − 5·25-s − 5·27-s − 3·29-s + 5·31-s + 8·37-s − 39-s + 3·41-s + 8·43-s + 9·47-s − 3·49-s − 6·51-s + 6·53-s + 2·57-s − 12·59-s + 14·61-s − 4·63-s + 8·67-s − 69-s − 15·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s − 2/3·9-s − 0.277·13-s − 1.45·17-s + 0.458·19-s + 0.436·21-s − 0.208·23-s − 25-s − 0.962·27-s − 0.557·29-s + 0.898·31-s + 1.31·37-s − 0.160·39-s + 0.468·41-s + 1.21·43-s + 1.31·47-s − 3/7·49-s − 0.840·51-s + 0.824·53-s + 0.264·57-s − 1.56·59-s + 1.79·61-s − 0.503·63-s + 0.977·67-s − 0.120·69-s − 1.78·71-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(92\)    =    \(2^{2} \cdot 23\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{92} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 92,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.136797478$
$L(\frac12)$  $\approx$  $1.136797478$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−19.99188837175700, −19.08345915727022, −17.83176396371620, −17.27251461743698, −15.89755216871646, −14.95280394700414, −14.10311958268329, −13.25550534507955, −11.80125427978392, −10.99981469295489, −9.514891056616410, −8.504802095603264, −7.498530944676989, −5.849673516902698, −4.292631118704495, −2.436522214885362, 2.436522214885362, 4.292631118704495, 5.849673516902698, 7.498530944676989, 8.504802095603264, 9.514891056616410, 10.99981469295489, 11.80125427978392, 13.25550534507955, 14.10311958268329, 14.95280394700414, 15.89755216871646, 17.27251461743698, 17.83176396371620, 19.08345915727022, 19.99188837175700

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