Properties

Label 1-175-175.12-r0-0-0
Degree $1$
Conductor $175$
Sign $0.445 + 0.895i$
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  + (0.743 + 0.669i)2-s + (0.994 − 0.104i)3-s + (0.104 + 0.994i)4-s + (0.809 + 0.587i)6-s + (−0.587 + 0.809i)8-s + (0.978 − 0.207i)9-s + (−0.978 − 0.207i)11-s + (0.207 + 0.978i)12-s + (0.951 + 0.309i)13-s + (−0.978 + 0.207i)16-s + (−0.406 − 0.913i)17-s + (0.866 + 0.5i)18-s + (−0.104 + 0.994i)19-s + (−0.587 − 0.809i)22-s + (−0.743 − 0.669i)23-s + (−0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (0.743 + 0.669i)2-s + (0.994 − 0.104i)3-s + (0.104 + 0.994i)4-s + (0.809 + 0.587i)6-s + (−0.587 + 0.809i)8-s + (0.978 − 0.207i)9-s + (−0.978 − 0.207i)11-s + (0.207 + 0.978i)12-s + (0.951 + 0.309i)13-s + (−0.978 + 0.207i)16-s + (−0.406 − 0.913i)17-s + (0.866 + 0.5i)18-s + (−0.104 + 0.994i)19-s + (−0.587 − 0.809i)22-s + (−0.743 − 0.669i)23-s + (−0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.445 + 0.895i$
Analytic conductor: \(0.812696\)
Root analytic conductor: \(0.812696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (0:\ ),\ 0.445 + 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.845771744 + 1.143038993i\)
\(L(\frac12)\) \(\approx\) \(1.845771744 + 1.143038993i\)
\(L(1)\) \(\approx\) \(1.750453031 + 0.7413313172i\)
\(L(1)\) \(\approx\) \(1.750453031 + 0.7413313172i\)

Euler product

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

−27.45556408862530962468525360806, −26.107512458272754072216854254467, −25.47228796617090491529451467414, −24.06953883453766018196031477361, −23.61130657333404339215875291935, −22.165484751016710406384439195109, −21.38344616511976111832265324314, −20.50642922870622090659128869576, −19.80538567251055963153603794115, −18.81531774813362672960364427645, −17.87110518109160047653441768335, −15.80831494596391096752528780783, −15.376308376277459809896250836, −14.18375185466297198512120196499, −13.288884114948197313604520952993, −12.66371901566048832859578799293, −11.081967054909351854230659937287, −10.25705593051692915654266476815, −9.10249538781686882563769435247, −7.93699434596559802193488428819, −6.46449567762989979978249426966, −5.04409433963575940107059952498, −3.87119072197319600893865217552, −2.84457730803590581904458264639, −1.66288615295180466083556372493, 2.212458104798647861463409184426, 3.385482658648017362371428634697, 4.45728136348529567181398516757, 5.86830684045937474640779986863, 7.09990344803579226045394857982, 8.10503883968373651603914880623, 8.90238484553561633652279238103, 10.43399257413620105495892382375, 11.94921926433969065463768102181, 13.07432580005583891540777248456, 13.79332207558468342481241902721, 14.61908732616375739999272345115, 15.80210504374703456387833408559, 16.268252162760589050589815917235, 17.97779448691120390198089605027, 18.68992913251605618765674423039, 20.23757104002346882862763656689, 20.88720057406349198385882307986, 21.73770279637264248231401038438, 23.04510122555937056112203756074, 23.82796200543119048009231326977, 24.832383537482092242665993494301, 25.52435187251510136150703111017, 26.43056392043810269881626836132, 27.090821942839933163122975612645

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