# Properties

 Label 2-1441-1441.916-c0-0-3 Degree $2$ Conductor $1441$ Sign $-0.887 + 0.460i$ Analytic cond. $0.719152$ Root an. cond. $0.848028$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.613 − 1.88i)3-s + (0.309 − 0.951i)4-s + (1.03 + 0.749i)5-s + (0.598 − 1.84i)7-s + (−2.37 + 1.72i)9-s + (−0.425 + 0.904i)11-s − 1.98·12-s + (0.303 − 0.220i)13-s + (0.781 − 2.40i)15-s + (−0.809 − 0.587i)16-s + (1.03 − 0.749i)20-s − 3.84·21-s + (0.193 + 0.594i)25-s + (3.10 + 2.25i)27-s + (−1.56 − 1.13i)28-s + ⋯
 L(s)  = 1 + (−0.613 − 1.88i)3-s + (0.309 − 0.951i)4-s + (1.03 + 0.749i)5-s + (0.598 − 1.84i)7-s + (−2.37 + 1.72i)9-s + (−0.425 + 0.904i)11-s − 1.98·12-s + (0.303 − 0.220i)13-s + (0.781 − 2.40i)15-s + (−0.809 − 0.587i)16-s + (1.03 − 0.749i)20-s − 3.84·21-s + (0.193 + 0.594i)25-s + (3.10 + 2.25i)27-s + (−1.56 − 1.13i)28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1441$$    =    $$11 \cdot 131$$ Sign: $-0.887 + 0.460i$ 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: $$(2,\ 1441,\ (\ :0),\ -0.887 + 0.460i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.111074086$$ $$L(\frac12)$$ $$\approx$$ $$1.111074086$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad11 $$1 + (0.425 - 0.904i)T$$
131 $$1 - T$$
good2 $$1 + (-0.309 + 0.951i)T^{2}$$
3 $$1 + (0.613 + 1.88i)T + (-0.809 + 0.587i)T^{2}$$
5 $$1 + (-1.03 - 0.749i)T + (0.309 + 0.951i)T^{2}$$
7 $$1 + (-0.598 + 1.84i)T + (-0.809 - 0.587i)T^{2}$$
13 $$1 + (-0.303 + 0.220i)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 + (-0.450 - 1.38i)T + (-0.809 + 0.587i)T^{2}$$
43 $$1 - 1.07T + 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 + (-0.0388 + 0.119i)T + (-0.809 - 0.587i)T^{2}$$
61 $$1 + (-1.03 - 0.749i)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

−9.710182705954584309205766246055, −8.161533452882893072531014436272, −7.36293984421791203779776273709, −6.96850885805695072825916070077, −6.29552528913398755438474902777, −5.58750834094308254453725581054, −4.63591013335201917488151613379, −2.66392466326180335927685706358, −1.77950587063949820314605008006, −1.03024471473904774924360263067, 2.29300372664594227228657079201, 3.18257998736867780663865013726, 4.27496882544134104259520021538, 5.25002080807157209692250930165, 5.62084046717936386133323286614, 6.29605222750459110748903808287, 8.175980446973584879937408003373, 8.849242634853497799885374254208, 9.059051040029955153473914798733, 9.914929781500349179600604120876