Properties

Label 1-175-175.61-r1-0-0
Degree $1$
Conductor $175$
Sign $0.668 - 0.743i$
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  + (−0.978 + 0.207i)2-s + (−0.913 + 0.406i)3-s + (0.913 − 0.406i)4-s + (0.809 − 0.587i)6-s + (−0.809 + 0.587i)8-s + (0.669 − 0.743i)9-s + (0.669 + 0.743i)11-s + (−0.669 + 0.743i)12-s + (−0.309 − 0.951i)13-s + (0.669 − 0.743i)16-s + (0.104 + 0.994i)17-s + (−0.5 + 0.866i)18-s + (−0.913 − 0.406i)19-s + (−0.809 − 0.587i)22-s + (−0.978 + 0.207i)23-s + (0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (−0.978 + 0.207i)2-s + (−0.913 + 0.406i)3-s + (0.913 − 0.406i)4-s + (0.809 − 0.587i)6-s + (−0.809 + 0.587i)8-s + (0.669 − 0.743i)9-s + (0.669 + 0.743i)11-s + (−0.669 + 0.743i)12-s + (−0.309 − 0.951i)13-s + (0.669 − 0.743i)16-s + (0.104 + 0.994i)17-s + (−0.5 + 0.866i)18-s + (−0.913 − 0.406i)19-s + (−0.809 − 0.587i)22-s + (−0.978 + 0.207i)23-s + (0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.668 - 0.743i$
Analytic conductor: \(18.8063\)
Root analytic conductor: \(18.8063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (1:\ ),\ 0.668 - 0.743i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5161030631 - 0.2301199811i\)
\(L(\frac12)\) \(\approx\) \(0.5161030631 - 0.2301199811i\)
\(L(1)\) \(\approx\) \(0.5160762281 + 0.04504407880i\)
\(L(1)\) \(\approx\) \(0.5160762281 + 0.04504407880i\)

Euler product

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

−27.50562262072588644185762829847, −26.62026052381406874375025804902, −25.4313281903320767616708118695, −24.45284304371979746492842057954, −23.78425377715457323877618501555, −22.3345362451038278202182212230, −21.59052188279388494518078301346, −20.39859661249086284167089006134, −19.12156704911221463149613701472, −18.66490858000194242790084988045, −17.55101741239240863975717657533, −16.63516894610641349667967061425, −16.154902599733263098953615336299, −14.53195228300308668742172878405, −13.09313066392175348163016428556, −11.83602825044026760751052181728, −11.399317873378141159248157323674, −10.19100443887369752313997343018, −9.11831163692058526699119541767, −7.83999517247967377506401059927, −6.76154488139667866207385516960, −5.90577686468146977651321103041, −4.14462357456967062983661561698, −2.29180842401211902222118421239, −0.98419802450263457613259754287, 0.39318826697475124992232343224, 1.94575661450135747360323141085, 3.937206070102657714431450214602, 5.46855590251439015188936729626, 6.429744461597909863293504187198, 7.52003072825342535442938187810, 8.86912296901611912790232957301, 10.01359896058654253116744154143, 10.652021909048248117537840640377, 11.82370072035675032603656353086, 12.70260597869900666111018109385, 14.75333775054437081158192917973, 15.39954216162939470574553241802, 16.50668082153797706160773295201, 17.43873532109323186251655175809, 17.84251594424845730174304290810, 19.22287058897338209494977417948, 20.12262530375853482780561064855, 21.2302124334349593509754863215, 22.2750474568413171118769952896, 23.34471620117810681445966736315, 24.24493979378000634412147111033, 25.3283210061120148833101688333, 26.24501371261082059196391937314, 27.32334813247989293783645239017

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