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 $0$

Origins

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Normalization:  

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 \( 1 + (-0.309 - 0.951i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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

Graph of the $Z$-function along the critical line