Properties

Label 2-1441-1.1-c1-0-72
Degree $2$
Conductor $1441$
Sign $1$
Analytic cond. $11.5064$
Root an. cond. $3.39211$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.30·2-s + 1.58·3-s + 3.30·4-s − 1.03·5-s + 3.65·6-s + 2.92·7-s + 3.00·8-s − 0.481·9-s − 2.39·10-s − 11-s + 5.24·12-s + 4.96·13-s + 6.73·14-s − 1.64·15-s + 0.308·16-s − 0.923·17-s − 1.10·18-s + 2.42·19-s − 3.43·20-s + 4.64·21-s − 2.30·22-s + 7.67·23-s + 4.76·24-s − 3.92·25-s + 11.4·26-s − 5.52·27-s + 9.66·28-s + ⋯
L(s)  = 1  + 1.62·2-s + 0.916·3-s + 1.65·4-s − 0.464·5-s + 1.49·6-s + 1.10·7-s + 1.06·8-s − 0.160·9-s − 0.756·10-s − 0.301·11-s + 1.51·12-s + 1.37·13-s + 1.80·14-s − 0.425·15-s + 0.0772·16-s − 0.223·17-s − 0.261·18-s + 0.556·19-s − 0.767·20-s + 1.01·21-s − 0.491·22-s + 1.60·23-s + 0.972·24-s − 0.784·25-s + 2.24·26-s − 1.06·27-s + 1.82·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $1$
Analytic conductor: \(11.5064\)
Root analytic conductor: \(3.39211\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1441,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.739220904\)
\(L(\frac12)\) \(\approx\) \(5.739220904\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + T \)
131 \( 1 - T \)
good2 \( 1 - 2.30T + 2T^{2} \)
3 \( 1 - 1.58T + 3T^{2} \)
5 \( 1 + 1.03T + 5T^{2} \)
7 \( 1 - 2.92T + 7T^{2} \)
13 \( 1 - 4.96T + 13T^{2} \)
17 \( 1 + 0.923T + 17T^{2} \)
19 \( 1 - 2.42T + 19T^{2} \)
23 \( 1 - 7.67T + 23T^{2} \)
29 \( 1 - 0.972T + 29T^{2} \)
31 \( 1 + 7.30T + 31T^{2} \)
37 \( 1 + 5.68T + 37T^{2} \)
41 \( 1 - 7.94T + 41T^{2} \)
43 \( 1 - 7.89T + 43T^{2} \)
47 \( 1 + 6.51T + 47T^{2} \)
53 \( 1 + 6.14T + 53T^{2} \)
59 \( 1 - 4.98T + 59T^{2} \)
61 \( 1 - 7.60T + 61T^{2} \)
67 \( 1 + 13.1T + 67T^{2} \)
71 \( 1 + 0.558T + 71T^{2} \)
73 \( 1 + 9.66T + 73T^{2} \)
79 \( 1 + 5.03T + 79T^{2} \)
83 \( 1 - 6.86T + 83T^{2} \)
89 \( 1 - 0.308T + 89T^{2} \)
97 \( 1 + 6.16T + 97T^{2} \)
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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

−9.262472296524611205887440345204, −8.612220578217796485228532883988, −7.81196866989769748289960177955, −7.09508367172804464199630004083, −5.90228082796799574207285365912, −5.25644005568198635050866213191, −4.30438892067286142020471296204, −3.56337095633470232823965749599, −2.82268731749055668678857498951, −1.66322937452274652005312330577, 1.66322937452274652005312330577, 2.82268731749055668678857498951, 3.56337095633470232823965749599, 4.30438892067286142020471296204, 5.25644005568198635050866213191, 5.90228082796799574207285365912, 7.09508367172804464199630004083, 7.81196866989769748289960177955, 8.612220578217796485228532883988, 9.262472296524611205887440345204

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