Properties

Degree 2
Conductor $ 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  − 2-s + 4-s + 4·5-s − 4·7-s − 8-s − 3·9-s − 4·10-s + 2·11-s − 2·13-s + 4·14-s + 16-s − 2·17-s + 3·18-s − 2·19-s + 4·20-s − 2·22-s + 23-s + 11·25-s + 2·26-s − 4·28-s + 2·29-s − 32-s + 2·34-s − 16·35-s − 3·36-s − 4·37-s + 2·38-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.78·5-s − 1.51·7-s − 0.353·8-s − 9-s − 1.26·10-s + 0.603·11-s − 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.707·18-s − 0.458·19-s + 0.894·20-s − 0.426·22-s + 0.208·23-s + 11/5·25-s + 0.392·26-s − 0.755·28-s + 0.371·29-s − 0.176·32-s + 0.342·34-s − 2.70·35-s − 1/2·36-s − 0.657·37-s + 0.324·38-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(46\)    =    \(2 \cdot 23\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{46} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 46,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6609041113$
$L(\frac12)$  $\approx$  $0.6609041113$
$L(\frac{3}{2})$   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 \( 1 + T \)
23 \( 1 - T \)
good3 \( 1 + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−19.85839284969429, −19.08356444645541, −17.67224309142049, −17.15745481995051, −16.21405768024039, −14.59615890156611, −13.53832745679506, −12.42275899419422, −10.68153496695016, −9.594811427439618, −8.971935452602448, −6.694419136417329, −5.844324179597279, −2.632327374745522, 2.632327374745522, 5.844324179597279, 6.694419136417329, 8.971935452602448, 9.594811427439618, 10.68153496695016, 12.42275899419422, 13.53832745679506, 14.59615890156611, 16.21405768024039, 17.15745481995051, 17.67224309142049, 19.08356444645541, 19.85839284969429

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