Properties

Degree 2
Conductor $ 2 \cdot 3 \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  − 2-s − 3-s + 4-s − 2·5-s + 6-s − 2·7-s − 8-s + 9-s + 2·10-s − 6·11-s − 12-s − 2·13-s + 2·14-s + 2·15-s + 16-s − 18-s − 2·20-s + 2·21-s + 6·22-s − 23-s + 24-s − 25-s + 2·26-s − 27-s − 2·28-s + 6·29-s − 2·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.80·11-s − 0.288·12-s − 0.554·13-s + 0.534·14-s + 0.516·15-s + 1/4·16-s − 0.235·18-s − 0.447·20-s + 0.436·21-s + 1.27·22-s − 0.208·23-s + 0.204·24-s − 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.377·28-s + 1.11·29-s − 0.365·30-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  =  1
Selberg data  =  $(2,\ 138,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;23\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]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 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−19.72906472145261, −19.18990879416691, −18.27068633064262, −17.52064588347563, −16.28460345242402, −15.89318934864779, −15.08969271474470, −13.42017203346058, −12.44601657534846, −11.61548664407487, −10.48952175513056, −9.832860358169758, −8.259124288211169, −7.489547860324460, −6.252505699268443, −4.785705824279448, −2.916312299383100, 0, 2.916312299383100, 4.785705824279448, 6.252505699268443, 7.489547860324460, 8.259124288211169, 9.832860358169758, 10.48952175513056, 11.61548664407487, 12.44601657534846, 13.42017203346058, 15.08969271474470, 15.89318934864779, 16.28460345242402, 17.52064588347563, 18.27068633064262, 19.18990879416691, 19.72906472145261

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