Properties

Label 2-1441-1.1-c1-0-42
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  − 2.09·2-s + 1.02·3-s + 2.36·4-s + 3.67·5-s − 2.14·6-s + 2.35·7-s − 0.772·8-s − 1.94·9-s − 7.68·10-s + 11-s + 2.42·12-s + 0.397·13-s − 4.92·14-s + 3.76·15-s − 3.12·16-s − 1.34·17-s + 4.07·18-s + 4.83·19-s + 8.70·20-s + 2.41·21-s − 2.09·22-s − 2.43·23-s − 0.791·24-s + 8.51·25-s − 0.829·26-s − 5.07·27-s + 5.58·28-s + ⋯
L(s)  = 1  − 1.47·2-s + 0.591·3-s + 1.18·4-s + 1.64·5-s − 0.874·6-s + 0.891·7-s − 0.273·8-s − 0.649·9-s − 2.42·10-s + 0.301·11-s + 0.701·12-s + 0.110·13-s − 1.31·14-s + 0.972·15-s − 0.781·16-s − 0.327·17-s + 0.960·18-s + 1.11·19-s + 1.94·20-s + 0.527·21-s − 0.445·22-s − 0.507·23-s − 0.161·24-s + 1.70·25-s − 0.162·26-s − 0.976·27-s + 1.05·28-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

\(L(1)\) \(\approx\) \(1.480632756\)
\(L(\frac12)\) \(\approx\) \(1.480632756\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.09T + 2T^{2} \)
3 \( 1 - 1.02T + 3T^{2} \)
5 \( 1 - 3.67T + 5T^{2} \)
7 \( 1 - 2.35T + 7T^{2} \)
13 \( 1 - 0.397T + 13T^{2} \)
17 \( 1 + 1.34T + 17T^{2} \)
19 \( 1 - 4.83T + 19T^{2} \)
23 \( 1 + 2.43T + 23T^{2} \)
29 \( 1 - 7.45T + 29T^{2} \)
31 \( 1 + 2.84T + 31T^{2} \)
37 \( 1 - 3.55T + 37T^{2} \)
41 \( 1 - 4.97T + 41T^{2} \)
43 \( 1 - 3.95T + 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 + 4.48T + 53T^{2} \)
59 \( 1 + 10.1T + 59T^{2} \)
61 \( 1 + 9.66T + 61T^{2} \)
67 \( 1 + 2.68T + 67T^{2} \)
71 \( 1 + 1.35T + 71T^{2} \)
73 \( 1 - 2.00T + 73T^{2} \)
79 \( 1 - 7.47T + 79T^{2} \)
83 \( 1 + 2.98T + 83T^{2} \)
89 \( 1 + 3.76T + 89T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−9.240754732864233683314083759133, −9.036963028730054877090993502074, −8.118419130885524194596787936364, −7.49642376080899259823826568448, −6.37712591620939234881916617015, −5.63826846487616276701554679045, −4.56219738448514130683969981323, −2.85073049964112147158837405463, −2.03476333045085613252670757819, −1.16448818337645602186047679856, 1.16448818337645602186047679856, 2.03476333045085613252670757819, 2.85073049964112147158837405463, 4.56219738448514130683969981323, 5.63826846487616276701554679045, 6.37712591620939234881916617015, 7.49642376080899259823826568448, 8.118419130885524194596787936364, 9.036963028730054877090993502074, 9.240754732864233683314083759133

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