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Properties

Label 1-175-175.54-r1-0-0
Degree $1$
Conductor $175$
Sign $-0.388 - 0.921i$
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  + (0.104 − 0.994i)2-s + (−0.978 − 0.207i)3-s + (−0.978 − 0.207i)4-s + (−0.309 + 0.951i)6-s + (−0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (0.913 − 0.406i)11-s + (0.913 + 0.406i)12-s + (−0.809 + 0.587i)13-s + (0.913 + 0.406i)16-s + (0.669 + 0.743i)17-s + (0.5 − 0.866i)18-s + (0.978 − 0.207i)19-s + (−0.309 − 0.951i)22-s + (0.104 − 0.994i)23-s + (0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (0.104 − 0.994i)2-s + (−0.978 − 0.207i)3-s + (−0.978 − 0.207i)4-s + (−0.309 + 0.951i)6-s + (−0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (0.913 − 0.406i)11-s + (0.913 + 0.406i)12-s + (−0.809 + 0.587i)13-s + (0.913 + 0.406i)16-s + (0.669 + 0.743i)17-s + (0.5 − 0.866i)18-s + (0.978 − 0.207i)19-s + (−0.309 − 0.951i)22-s + (0.104 − 0.994i)23-s + (0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.388 - 0.921i$
Analytic conductor: \(18.8063\)
Root analytic conductor: \(18.8063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (1:\ ),\ -0.388 - 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6408245378 - 0.9656563767i\)
\(L(\frac12)\) \(\approx\) \(0.6408245378 - 0.9656563767i\)
\(L(1)\) \(\approx\) \(0.6843793783 - 0.4614661394i\)
\(L(1)\) \(\approx\) \(0.6843793783 - 0.4614661394i\)

Euler product

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

−27.338654559099707973568548357078, −26.769692817824609734279903562912, −25.30577835726120420953913810466, −24.66825919552778302913651996642, −23.61442804401579731563817912234, −22.69062862268920662579308581289, −22.21746400195263231889260758021, −21.07839837649839993097936327966, −19.55435629364718219610639514248, −18.25565412595590068360512327730, −17.5097086520894257382942102932, −16.73193560968362100144117338893, −15.80558140682091713493513249134, −14.87421420659447379292389787960, −13.774967598015436172195676929832, −12.45364407555214638015147779495, −11.712640160306578983800133461119, −10.03891454916207922664669572172, −9.366334141003061673015697684457, −7.66619403348458555716760348847, −6.86606124446776009840993934313, −5.63306673981735830316593653560, −4.87024299763742503397300931044, −3.56141588436997039435408074869, −0.95378570221037474978470123060, 0.65015434292404332810958888893, 1.89890157816091942696394822316, 3.62704001019302029202415051749, 4.806908226701130301516198528259, 5.882607208873193123630091873878, 7.23057195659660161535006681318, 8.84363229621183182695277615427, 9.96831881498723940097177547664, 10.92858430129044319129838345487, 11.93081243085287859243057537787, 12.47977190740351363339972710025, 13.770407689073853321607177710933, 14.75427719290869731148286558246, 16.48367160383432176671749904875, 17.18624559864900232381523684804, 18.28631435188751758115453272485, 19.09797525966795037654014349111, 20.02217892003999177805520847029, 21.36837798641551642645425903396, 22.048277869421071314658310959568, 22.76935275439530726240730175839, 23.87338810117412444906974582569, 24.59471055946933885510985557468, 26.326621506314405294162992286172, 27.2625879144536942188781433977

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