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

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