# Properties

 Label 2-1148-41.9-c1-0-3 Degree $2$ Conductor $1148$ Sign $-0.783 - 0.621i$ Analytic cond. $9.16682$ Root an. cond. $3.02767$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.65 + 1.65i)3-s + 2.95i·5-s + (−0.707 − 0.707i)7-s + 2.45i·9-s + (0.583 + 0.583i)11-s + (−0.304 − 0.304i)13-s + (−4.88 + 4.88i)15-s + (−4.59 + 4.59i)17-s + (−2.24 + 2.24i)19-s − 2.33i·21-s − 2.62·23-s − 3.72·25-s + (0.893 − 0.893i)27-s + (3.12 + 3.12i)29-s + 3.25·31-s + ⋯
 L(s)  = 1 + (0.953 + 0.953i)3-s + 1.32i·5-s + (−0.267 − 0.267i)7-s + 0.819i·9-s + (0.175 + 0.175i)11-s + (−0.0843 − 0.0843i)13-s + (−1.26 + 1.26i)15-s + (−1.11 + 1.11i)17-s + (−0.514 + 0.514i)19-s − 0.509i·21-s − 0.546·23-s − 0.745·25-s + (0.171 − 0.171i)27-s + (0.580 + 0.580i)29-s + 0.584·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 - 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1148$$    =    $$2^{2} \cdot 7 \cdot 41$$ Sign: $-0.783 - 0.621i$ Analytic conductor: $$9.16682$$ Root analytic conductor: $$3.02767$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1148} (337, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1148,\ (\ :1/2),\ -0.783 - 0.621i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.870070267$$ $$L(\frac12)$$ $$\approx$$ $$1.870070267$$ $$L(\frac{3}{2})$$ 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 + (6.24 + 1.41i)T$$
good3 $$1 + (-1.65 - 1.65i)T + 3iT^{2}$$
5 $$1 - 2.95iT - 5T^{2}$$
11 $$1 + (-0.583 - 0.583i)T + 11iT^{2}$$
13 $$1 + (0.304 + 0.304i)T + 13iT^{2}$$
17 $$1 + (4.59 - 4.59i)T - 17iT^{2}$$
19 $$1 + (2.24 - 2.24i)T - 19iT^{2}$$
23 $$1 + 2.62T + 23T^{2}$$
29 $$1 + (-3.12 - 3.12i)T + 29iT^{2}$$
31 $$1 - 3.25T + 31T^{2}$$
37 $$1 - 9.55T + 37T^{2}$$
43 $$1 + 4.05iT - 43T^{2}$$
47 $$1 + (7.44 - 7.44i)T - 47iT^{2}$$
53 $$1 + (-0.577 - 0.577i)T + 53iT^{2}$$
59 $$1 - 5.94T + 59T^{2}$$
61 $$1 - 5.32iT - 61T^{2}$$
67 $$1 + (-6.10 + 6.10i)T - 67iT^{2}$$
71 $$1 + (7.75 + 7.75i)T + 71iT^{2}$$
73 $$1 + 5.39iT - 73T^{2}$$
79 $$1 + (-3.82 - 3.82i)T + 79iT^{2}$$
83 $$1 - 4.27T + 83T^{2}$$
89 $$1 + (-1.40 - 1.40i)T + 89iT^{2}$$
97 $$1 + (-5.46 + 5.46i)T - 97iT^{2}$$
show less
$$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

−10.25941128428107528501166877526, −9.407788274300175532907813646136, −8.540559704663345102852697862400, −7.82519915545490943594821339232, −6.68535373621291214390263452489, −6.19826716319602008695424409098, −4.59315434113661938464964780884, −3.84708533321525448780821721901, −3.08542284286396475075170397981, −2.14229150926251209393787920606, 0.70158030636500439279624008071, 2.00653623305084135358882638259, 2.86227405261891288363898728395, 4.29601716805904282846950532697, 5.06513907128038743789220526053, 6.33119068415534897637188242820, 7.03775131119282690027139540720, 8.141705884036513645820980055137, 8.521180835985238033136714509334, 9.222507364332531209254588212721