Properties

Label 2-1441-1441.654-c0-0-2
Degree $2$
Conductor $1441$
Sign $0.410 - 0.911i$
Analytic cond. $0.719152$
Root an. cond. $0.848028$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.50 + 1.09i)3-s + (−0.809 + 0.587i)4-s + (0.541 − 1.66i)5-s + (−1.17 + 0.856i)7-s + (0.759 + 2.33i)9-s + (0.968 − 0.248i)11-s − 1.85·12-s + (0.331 + 1.01i)13-s + (2.63 − 1.91i)15-s + (0.309 − 0.951i)16-s + (0.541 + 1.66i)20-s − 2.71·21-s + (−1.67 − 1.21i)25-s + (−0.837 + 2.57i)27-s + (0.450 − 1.38i)28-s + ⋯
L(s)  = 1  + (1.50 + 1.09i)3-s + (−0.809 + 0.587i)4-s + (0.541 − 1.66i)5-s + (−1.17 + 0.856i)7-s + (0.759 + 2.33i)9-s + (0.968 − 0.248i)11-s − 1.85·12-s + (0.331 + 1.01i)13-s + (2.63 − 1.91i)15-s + (0.309 − 0.951i)16-s + (0.541 + 1.66i)20-s − 2.71·21-s + (−1.67 − 1.21i)25-s + (−0.837 + 2.57i)27-s + (0.450 − 1.38i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.410 - 0.911i$
Analytic conductor: \(0.719152\)
Root analytic conductor: \(0.848028\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (654, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1441,\ (\ :0),\ 0.410 - 0.911i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.526971278\)
\(L(\frac12)\) \(\approx\) \(1.526971278\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.968 + 0.248i)T \)
131 \( 1 - T \)
good2 \( 1 + (0.809 - 0.587i)T^{2} \)
3 \( 1 + (-1.50 - 1.09i)T + (0.309 + 0.951i)T^{2} \)
5 \( 1 + (-0.541 + 1.66i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (1.17 - 0.856i)T + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.331 - 1.01i)T + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (-1.03 - 0.749i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 + 1.98T + T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.303 + 0.220i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.541 + 1.66i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 - 0.618T + T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.506189150252004917652348457947, −9.176045854378435001212632230568, −8.538935860702995802663917376395, −8.164687320162107103046232722163, −6.58318961177792916694845937564, −5.36928495117008190412792716123, −4.59051035927310234079420535654, −3.89504257867996471024099131455, −3.14851880120488134429331903075, −1.86122459171333006384042479221, 1.25191162432772337907878948151, 2.55272605768023836687961193669, 3.40234619243797604288240809360, 3.91452509468985041480615621119, 5.91679963132776309889057338376, 6.55512206989556622610799130480, 7.07347817931145150721025809375, 7.85175125151316321933603824039, 8.899123977006944040242698844294, 9.566585568040747361012529651331

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