Properties

Label 2-1441-1.1-c1-0-29
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  − 2.51·2-s − 0.0469·3-s + 4.33·4-s + 2.78·5-s + 0.118·6-s + 3.17·7-s − 5.88·8-s − 2.99·9-s − 7.00·10-s − 11-s − 0.203·12-s − 3.12·13-s − 7.98·14-s − 0.130·15-s + 6.14·16-s + 1.69·17-s + 7.54·18-s − 6.80·19-s + 12.0·20-s − 0.149·21-s + 2.51·22-s + 5.59·23-s + 0.276·24-s + 2.73·25-s + 7.85·26-s + 0.281·27-s + 13.7·28-s + ⋯
L(s)  = 1  − 1.78·2-s − 0.0271·3-s + 2.16·4-s + 1.24·5-s + 0.0482·6-s + 1.19·7-s − 2.08·8-s − 0.999·9-s − 2.21·10-s − 0.301·11-s − 0.0588·12-s − 0.865·13-s − 2.13·14-s − 0.0337·15-s + 1.53·16-s + 0.410·17-s + 1.77·18-s − 1.56·19-s + 2.69·20-s − 0.0325·21-s + 0.536·22-s + 1.16·23-s + 0.0564·24-s + 0.547·25-s + 1.54·26-s + 0.0542·27-s + 2.60·28-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: $\chi_{1441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1441,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9217166945\)
\(L(\frac12)\) \(\approx\) \(0.9217166945\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.51T + 2T^{2} \)
3 \( 1 + 0.0469T + 3T^{2} \)
5 \( 1 - 2.78T + 5T^{2} \)
7 \( 1 - 3.17T + 7T^{2} \)
13 \( 1 + 3.12T + 13T^{2} \)
17 \( 1 - 1.69T + 17T^{2} \)
19 \( 1 + 6.80T + 19T^{2} \)
23 \( 1 - 5.59T + 23T^{2} \)
29 \( 1 - 0.262T + 29T^{2} \)
31 \( 1 - 10.5T + 31T^{2} \)
37 \( 1 - 0.913T + 37T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 - 0.898T + 43T^{2} \)
47 \( 1 + 1.97T + 47T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 - 1.45T + 59T^{2} \)
61 \( 1 - 9.70T + 61T^{2} \)
67 \( 1 + 5.64T + 67T^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 - 5.38T + 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 + 7.57T + 83T^{2} \)
89 \( 1 - 14.1T + 89T^{2} \)
97 \( 1 + 10.3T + 97T^{2} \)
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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

−9.523159884249851120267099538531, −8.623990236961428241449198951196, −8.293532841894935451063002738688, −7.37827652521128336085864139506, −6.45454599331058968924620928317, −5.66366507193348300137772446618, −4.72027707375781101971543309257, −2.59925645380564837813741974936, −2.18004416913877972526758529785, −0.905368934988066937914254489066, 0.905368934988066937914254489066, 2.18004416913877972526758529785, 2.59925645380564837813741974936, 4.72027707375781101971543309257, 5.66366507193348300137772446618, 6.45454599331058968924620928317, 7.37827652521128336085864139506, 8.293532841894935451063002738688, 8.623990236961428241449198951196, 9.523159884249851120267099538531

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