Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 11 \cdot 23^{2} $
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 − 2·7-s + 8-s − 11-s − 4·13-s − 2·14-s + 16-s − 6·17-s + 4·19-s − 22-s − 5·25-s − 4·26-s − 2·28-s − 6·29-s + 8·31-s + 32-s − 6·34-s + 10·37-s + 4·38-s − 6·41-s − 8·43-s − 44-s + 6·47-s − 3·49-s − 5·50-s − 4·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s − 0.301·11-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s − 0.213·22-s − 25-s − 0.784·26-s − 0.377·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s − 1.02·34-s + 1.64·37-s + 0.648·38-s − 0.937·41-s − 1.21·43-s − 0.150·44-s + 0.875·47-s − 3/7·49-s − 0.707·50-s − 0.554·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104742 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104742 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(104742\)    =    \(2 \cdot 3^{2} \cdot 11 \cdot 23^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{104742} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 104742,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.151502420$
$L(\frac12)$  $\approx$  $1.151502420$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;23\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
11 \( 1 + T \)
23 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.71526802454961, −13.20999999333811, −12.87053868000510, −12.28137653618769, −11.82381323810037, −11.32643994624719, −11.01155663469963, −10.03657919598381, −9.908977030169332, −9.477582599962136, −8.715979269464900, −8.189657741190867, −7.453552548139540, −7.229696629896775, −6.531974945554586, −6.127254348178527, −5.532078231013295, −4.969400737871550, −4.394399515538868, −3.989449462765231, −3.113380050395691, −2.771138423366196, −2.159984409268160, −1.421397039625412, −0.2792167317276017, 0.2792167317276017, 1.421397039625412, 2.159984409268160, 2.771138423366196, 3.113380050395691, 3.989449462765231, 4.394399515538868, 4.969400737871550, 5.532078231013295, 6.127254348178527, 6.531974945554586, 7.229696629896775, 7.453552548139540, 8.189657741190867, 8.715979269464900, 9.477582599962136, 9.908977030169332, 10.03657919598381, 11.01155663469963, 11.32643994624719, 11.82381323810037, 12.28137653618769, 12.87053868000510, 13.20999999333811, 13.71526802454961

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