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Properties

Label 1-175-175.19-r1-0-0
Degree $1$
Conductor $175$
Sign $-0.784 - 0.620i$
Analytic cond. $18.8063$
Root an. cond. $18.8063$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.784 - 0.620i$
Analytic conductor: \(18.8063\)
Root analytic conductor: \(18.8063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (1:\ ),\ -0.784 - 0.620i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1004637897 - 0.2889982126i\)
\(L(\frac12)\) \(\approx\) \(0.1004637897 - 0.2889982126i\)
\(L(1)\) \(\approx\) \(0.6622471822 + 0.01695747811i\)
\(L(1)\) \(\approx\) \(0.6622471822 + 0.01695747811i\)

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

−27.47399310646479817755733776093, −26.23427974482727058578846596562, −25.906872663269893839529836090976, −24.76438307436439795216663453246, −24.09984686268694244927792490579, −23.21993255826872368483925528053, −21.62199022689452922356231201065, −20.22304883320421487486049189345, −19.767388188915387391724977982183, −18.79099043393486874705604660976, −17.789472448513546467625956724583, −17.17683727565314910377844806329, −15.64630636386789893701557167507, −14.89701090207785360050614445476, −13.885809835606259636585067013513, −12.58166574192578198564582829259, −11.51612758709638102142335963949, −10.05631194321302338474396411018, −9.15066211817786702640955517163, −8.16023040787768327018322423274, −7.11286468257903345417755955107, −6.42009429722095356766996004506, −4.630620602639435410161905241734, −2.57863215701008563990178579982, −1.65360713482098550453636170061, 0.12483836550105255285748878788, 2.155490030012670426826392939585, 3.17534119905223603484695905208, 4.4469471101859149506011138569, 6.215260577627942811819672286190, 7.850081788450687246400434668829, 8.477658445469221099995326100488, 9.62667422398358045678132365846, 10.43521099830069625794269994535, 11.39208065460706547454681423482, 12.738109045299667514364048945951, 13.9635557281069424162558847043, 15.2174258421286470318487067567, 16.0971044309881115438592849772, 16.96968335724449388094234864599, 18.12201706962177162113125144434, 19.326180845827483525289027901862, 19.845123734802096807562054034771, 20.95123378619839136599968616155, 21.62838143974617662350616518211, 22.60558245850418974051333283152, 24.498959840434345451069980362979, 25.02295979585645316776323792949, 26.32021974461704915107192414151, 26.75268167588948104807180287926

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