# Properties

 Label 2-1148-41.32-c1-0-10 Degree $2$ Conductor $1148$ Sign $0.496 - 0.867i$ Analytic cond. $9.16682$ Root an. cond. $3.02767$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−0.796 + 0.796i)3-s + 1.53i·5-s + (0.707 − 0.707i)7-s + 1.73i·9-s + (2.10 − 2.10i)11-s + (3.47 − 3.47i)13-s + (−1.22 − 1.22i)15-s + (1.03 + 1.03i)17-s + (2.78 + 2.78i)19-s + 1.12i·21-s − 4.78·23-s + 2.62·25-s + (−3.76 − 3.76i)27-s + (1.87 − 1.87i)29-s + 1.64·31-s + ⋯
 L(s)  = 1 + (−0.460 + 0.460i)3-s + 0.688i·5-s + (0.267 − 0.267i)7-s + 0.576i·9-s + (0.635 − 0.635i)11-s + (0.964 − 0.964i)13-s + (−0.316 − 0.316i)15-s + (0.250 + 0.250i)17-s + (0.638 + 0.638i)19-s + 0.245i·21-s − 0.998·23-s + 0.525·25-s + (−0.725 − 0.725i)27-s + (0.347 − 0.347i)29-s + 0.295·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 - 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.496 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1148$$    =    $$2^{2} \cdot 7 \cdot 41$$ Sign: $0.496 - 0.867i$ Analytic conductor: $$9.16682$$ Root analytic conductor: $$3.02767$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1148} (729, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1148,\ (\ :1/2),\ 0.496 - 0.867i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.542508321$$ $$L(\frac12)$$ $$\approx$$ $$1.542508321$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
7 $$1 + (-0.707 + 0.707i)T$$
41 $$1 + (5.27 - 3.63i)T$$
good3 $$1 + (0.796 - 0.796i)T - 3iT^{2}$$
5 $$1 - 1.53iT - 5T^{2}$$
11 $$1 + (-2.10 + 2.10i)T - 11iT^{2}$$
13 $$1 + (-3.47 + 3.47i)T - 13iT^{2}$$
17 $$1 + (-1.03 - 1.03i)T + 17iT^{2}$$
19 $$1 + (-2.78 - 2.78i)T + 19iT^{2}$$
23 $$1 + 4.78T + 23T^{2}$$
29 $$1 + (-1.87 + 1.87i)T - 29iT^{2}$$
31 $$1 - 1.64T + 31T^{2}$$
37 $$1 - 7.76T + 37T^{2}$$
43 $$1 - 7.65iT - 43T^{2}$$
47 $$1 + (-6.34 - 6.34i)T + 47iT^{2}$$
53 $$1 + (3.97 - 3.97i)T - 53iT^{2}$$
59 $$1 + 4.91T + 59T^{2}$$
61 $$1 - 14.5iT - 61T^{2}$$
67 $$1 + (11.3 + 11.3i)T + 67iT^{2}$$
71 $$1 + (-6.65 + 6.65i)T - 71iT^{2}$$
73 $$1 - 12.2iT - 73T^{2}$$
79 $$1 + (-2.18 + 2.18i)T - 79iT^{2}$$
83 $$1 - 10.0T + 83T^{2}$$
89 $$1 + (-11.2 + 11.2i)T - 89iT^{2}$$
97 $$1 + (-12.3 - 12.3i)T + 97iT^{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

−10.28023546258442135521477177366, −9.230582066323432119291135928242, −8.057772058652577406493575680824, −7.71795288039483311647125164196, −6.23801531296872444265627552945, −5.93687482017770804373080092655, −4.75038313852995140820498394211, −3.78808367667669906040120212885, −2.85737517136193751794053137044, −1.20808901918651639494344306610, 0.895294003973373486423654544764, 1.91938429446615480135859620482, 3.57227132695960166230536419436, 4.51717390683476195615332653169, 5.44696135057167139241418741729, 6.41449479039926866917222888896, 6.97239719524545817020822536558, 8.057939917975784243919326108732, 9.056694414803823983908875865451, 9.338282946765510271560722242386