# Properties

 Label 1-175-175.4-r0-0-0 Degree $1$ Conductor $175$ Sign $-0.186 + 0.982i$ Analytic cond. $0.812696$ Root an. cond. $0.812696$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.913 + 0.406i)2-s + (−0.669 + 0.743i)3-s + (0.669 − 0.743i)4-s + (0.309 − 0.951i)6-s + (−0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + (−0.104 + 0.994i)11-s + (0.104 + 0.994i)12-s + (0.809 − 0.587i)13-s + (−0.104 − 0.994i)16-s + (0.978 − 0.207i)17-s + (0.5 + 0.866i)18-s + (0.669 + 0.743i)19-s + (−0.309 − 0.951i)22-s + (−0.913 + 0.406i)23-s + (−0.5 − 0.866i)24-s + ⋯
 L(s)  = 1 + (−0.913 + 0.406i)2-s + (−0.669 + 0.743i)3-s + (0.669 − 0.743i)4-s + (0.309 − 0.951i)6-s + (−0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + (−0.104 + 0.994i)11-s + (0.104 + 0.994i)12-s + (0.809 − 0.587i)13-s + (−0.104 − 0.994i)16-s + (0.978 − 0.207i)17-s + (0.5 + 0.866i)18-s + (0.669 + 0.743i)19-s + (−0.309 − 0.951i)22-s + (−0.913 + 0.406i)23-s + (−0.5 − 0.866i)24-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$175$$    =    $$5^{2} \cdot 7$$ Sign: $-0.186 + 0.982i$ Analytic conductor: $$0.812696$$ Root analytic conductor: $$0.812696$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{175} (4, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 175,\ (0:\ ),\ -0.186 + 0.982i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.3663937768 + 0.4426490901i$$ $$L(\frac12)$$ $$\approx$$ $$0.3663937768 + 0.4426490901i$$ $$L(1)$$ $$\approx$$ $$0.5341786932 + 0.2789702322i$$ $$L(1)$$ $$\approx$$ $$0.5341786932 + 0.2789702322i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
7 $$1$$
good2 $$1 + (-0.913 + 0.406i)T$$
3 $$1 + (-0.669 + 0.743i)T$$
11 $$1 + (-0.104 + 0.994i)T$$
13 $$1 + (0.809 - 0.587i)T$$
17 $$1 + (0.978 - 0.207i)T$$
19 $$1 + (0.669 + 0.743i)T$$
23 $$1 + (-0.913 + 0.406i)T$$
29 $$1 + (0.309 + 0.951i)T$$
31 $$1 + (-0.978 + 0.207i)T$$
37 $$1 + (0.104 + 0.994i)T$$
41 $$1 + (-0.809 + 0.587i)T$$
43 $$1 - T$$
47 $$1 + (0.978 + 0.207i)T$$
53 $$1 + (-0.669 + 0.743i)T$$
59 $$1 + (0.913 + 0.406i)T$$
61 $$1 + (0.913 - 0.406i)T$$
67 $$1 + (0.978 - 0.207i)T$$
71 $$1 + (0.309 + 0.951i)T$$
73 $$1 + (0.104 - 0.994i)T$$
79 $$1 + (-0.978 - 0.207i)T$$
83 $$1 + (-0.309 + 0.951i)T$$
89 $$1 + (0.913 - 0.406i)T$$
97 $$1 + (-0.309 - 0.951i)T$$
show less
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−27.32360664389893743406457547464, −26.29596048229916674310804294552, −25.35377203812836987679114660137, −24.32583876599417469176628562888, −23.55301591072340945229208546990, −22.174955222095486371432504247125, −21.3621937424928694695801091853, −20.16179637599750818257655654388, −19.02429760542845403053326144224, −18.5420670975370629941929256708, −17.557057648876657321599190862530, −16.54326333401044815170726221965, −15.92405202049712753696354760950, −13.99848116767246512295151095831, −12.982315968775081905196907419407, −11.81797906993077135535782537058, −11.20304799231329006388686623618, −10.12099789466043311970881329856, −8.71954498589750014233411864402, −7.80205328278696179864112700019, −6.66710770638359094413527766590, −5.63651557428134597368678819293, −3.6443848951977584895607889137, −2.117375854949201683642130185729, −0.76970999019076527444793387568, 1.370531132035735975708970685898, 3.40152188083066647307876513463, 5.08362277021700828071641901804, 5.955219244378527453233100597614, 7.21382803268921574261022835807, 8.40451860255734334986093358664, 9.74692913017642993586407421880, 10.23334487127032749472516057858, 11.41018603423106410851340100156, 12.391838999414664587889683348594, 14.256776337452756951374912506419, 15.26454544915054426447339136218, 16.067654138513503574163782900569, 16.87434187697884987196946446474, 17.95224260609399354002262608870, 18.493807996876596311871551564077, 20.19747110826429243811400697403, 20.59752516340532199104654444874, 21.97371243500085308817504420293, 23.16334733191662685828238065229, 23.69209173648556811101173496364, 25.252323358924451686207763500693, 25.77457015538375428687021432061, 26.97199280831137544427256798104, 27.66973589334803363943053523315