Properties

Degree 2
Conductor $ 2^{2} \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  − 3·3-s − 2·5-s − 4·7-s + 6·9-s + 2·11-s − 5·13-s + 6·15-s + 4·17-s − 2·19-s + 12·21-s + 23-s − 25-s − 9·27-s − 7·29-s − 3·31-s − 6·33-s + 8·35-s + 2·37-s + 15·39-s − 9·41-s − 8·43-s − 12·45-s + 9·47-s + 9·49-s − 12·51-s + 2·53-s − 4·55-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.894·5-s − 1.51·7-s + 2·9-s + 0.603·11-s − 1.38·13-s + 1.54·15-s + 0.970·17-s − 0.458·19-s + 2.61·21-s + 0.208·23-s − 1/5·25-s − 1.73·27-s − 1.29·29-s − 0.538·31-s − 1.04·33-s + 1.35·35-s + 0.328·37-s + 2.40·39-s − 1.40·41-s − 1.21·43-s − 1.78·45-s + 1.31·47-s + 9/7·49-s − 1.68·51-s + 0.274·53-s − 0.539·55-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  =  1
Selberg data  =  $(2,\ 92,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + p T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
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 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−19.19206590632165, −18.57965138501728, −17.00910074010605, −16.82427123663198, −15.86466348526688, −14.89277282765627, −13.03943744344059, −12.21704194536702, −11.67692432657301, −10.38750929477444, −9.520156045863342, −7.412470652004118, −6.534832335614517, −5.321545291443746, −3.802065178928029, 0, 3.802065178928029, 5.321545291443746, 6.534832335614517, 7.412470652004118, 9.520156045863342, 10.38750929477444, 11.67692432657301, 12.21704194536702, 13.03943744344059, 14.89277282765627, 15.86466348526688, 16.82427123663198, 17.00910074010605, 18.57965138501728, 19.19206590632165

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