Properties

Label 2-1441-1441.916-c0-0-0
Degree $2$
Conductor $1441$
Sign $0.447 - 0.894i$
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  + (0.541 + 1.66i)3-s + (0.309 − 0.951i)4-s + (1.50 + 1.09i)5-s + (0.331 − 1.01i)7-s + (−1.67 + 1.21i)9-s + (−0.187 − 0.982i)11-s + 1.75·12-s + (−1.17 + 0.856i)13-s + (−1.00 + 3.09i)15-s + (−0.809 − 0.587i)16-s + (1.50 − 1.09i)20-s + 1.87·21-s + (0.759 + 2.33i)25-s + (−1.51 − 1.10i)27-s + (−0.866 − 0.629i)28-s + ⋯
L(s)  = 1  + (0.541 + 1.66i)3-s + (0.309 − 0.951i)4-s + (1.50 + 1.09i)5-s + (0.331 − 1.01i)7-s + (−1.67 + 1.21i)9-s + (−0.187 − 0.982i)11-s + 1.75·12-s + (−1.17 + 0.856i)13-s + (−1.00 + 3.09i)15-s + (−0.809 − 0.587i)16-s + (1.50 − 1.09i)20-s + 1.87·21-s + (0.759 + 2.33i)25-s + (−1.51 − 1.10i)27-s + (−0.866 − 0.629i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(0.719152\)
Root analytic conductor: \(0.848028\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (916, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1441,\ (\ :0),\ 0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.668696702\)
\(L(\frac12)\) \(\approx\) \(1.668696702\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.187 + 0.982i)T \)
131 \( 1 - T \)
good2 \( 1 + (-0.309 + 0.951i)T^{2} \)
3 \( 1 + (-0.541 - 1.66i)T + (-0.809 + 0.587i)T^{2} \)
5 \( 1 + (-1.50 - 1.09i)T + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (-0.331 + 1.01i)T + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (1.17 - 0.856i)T + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.613 + 1.88i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + 1.27T + T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.598 + 1.84i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-1.50 - 1.09i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + 1.61T + T^{2} \)
97 \( 1 + (-0.309 + 0.951i)T^{2} \)
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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

−9.938292430671806831416739685070, −9.502326350821678508280091807148, −8.647529141222961817914262946548, −7.24387849318737470156499799414, −6.53284635600522783307680798625, −5.50034725476555119192310463714, −5.00261733748109193600448407009, −3.86119133159626993095609393741, −2.82337589111669778181486003975, −1.96735912859774328563135863310, 1.59040690498559854027892320784, 2.28482034253142266103178364332, 2.83792465212348602887162471077, 4.80703768568097760864930159614, 5.54536580723250139552932338256, 6.48407572596504224048809473524, 7.21253620110141079416704797676, 8.175996030978487469734371418785, 8.436267539562733958427163119063, 9.371345011806597915494488993907

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