Properties

Degree 2
Conductor $ 2 \cdot 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  − 2-s + 3-s + 4-s − 6-s + 2·7-s − 8-s + 9-s + 12-s + 2·13-s − 2·14-s + 16-s − 18-s + 2·19-s + 2·21-s − 23-s − 24-s − 5·25-s − 2·26-s + 27-s + 2·28-s − 6·29-s − 4·31-s − 32-s + 36-s − 10·37-s − 2·38-s + 2·39-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.554·13-s − 0.534·14-s + 1/4·16-s − 0.235·18-s + 0.458·19-s + 0.436·21-s − 0.208·23-s − 0.204·24-s − 25-s − 0.392·26-s + 0.192·27-s + 0.377·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 1/6·36-s − 1.64·37-s − 0.324·38-s + 0.320·39-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(138\)    =    \(2 \cdot 3 \cdot 23\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{138} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 138,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.020067807$
$L(\frac12)$  $\approx$  $1.020067807$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;23\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;23\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 - T \)
23 \( 1 + T \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−19.59705698979389, −18.66372116583871, −17.97827464810006, −17.07701709125758, −16.00961115408310, −15.18929069599152, −14.24122148751627, −13.29718048196762, −11.97123091701779, −11.05538094544138, −9.978255855064832, −8.909988188498167, −8.056843455950655, −7.066136426809350, −5.481936774488981, −3.699597488735358, −1.859313160109573, 1.859313160109573, 3.699597488735358, 5.481936774488981, 7.066136426809350, 8.056843455950655, 8.909988188498167, 9.978255855064832, 11.05538094544138, 11.97123091701779, 13.29718048196762, 14.24122148751627, 15.18929069599152, 16.00961115408310, 17.07701709125758, 17.97827464810006, 18.66372116583871, 19.59705698979389

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