Properties

Label 1-175-175.172-r1-0-0
Degree $1$
Conductor $175$
Sign $0.186 - 0.982i$
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  + (0.406 + 0.913i)2-s + (−0.743 − 0.669i)3-s + (−0.669 + 0.743i)4-s + (0.309 − 0.951i)6-s + (−0.951 − 0.309i)8-s + (0.104 + 0.994i)9-s + (−0.104 + 0.994i)11-s + (0.994 − 0.104i)12-s + (0.587 + 0.809i)13-s + (−0.104 − 0.994i)16-s + (−0.207 − 0.978i)17-s + (−0.866 + 0.5i)18-s + (−0.669 − 0.743i)19-s + (−0.951 + 0.309i)22-s + (−0.406 − 0.913i)23-s + (0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (0.406 + 0.913i)2-s + (−0.743 − 0.669i)3-s + (−0.669 + 0.743i)4-s + (0.309 − 0.951i)6-s + (−0.951 − 0.309i)8-s + (0.104 + 0.994i)9-s + (−0.104 + 0.994i)11-s + (0.994 − 0.104i)12-s + (0.587 + 0.809i)13-s + (−0.104 − 0.994i)16-s + (−0.207 − 0.978i)17-s + (−0.866 + 0.5i)18-s + (−0.669 − 0.743i)19-s + (−0.951 + 0.309i)22-s + (−0.406 − 0.913i)23-s + (0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.186 - 0.982i$
Analytic conductor: \(18.8063\)
Root analytic conductor: \(18.8063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (1:\ ),\ 0.186 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3992460186 - 0.3304677676i\)
\(L(\frac12)\) \(\approx\) \(0.3992460186 - 0.3304677676i\)
\(L(1)\) \(\approx\) \(0.7489038794 + 0.2031958430i\)
\(L(1)\) \(\approx\) \(0.7489038794 + 0.2031958430i\)

Euler product

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

−27.59288213162954188626837846148, −26.933944252647383569401135388509, −25.6516128878224614197732957518, −23.96740872746370377569288042831, −23.458132212377948719026359736263, −22.32455599560017906973984701311, −21.68145551086934497752420879156, −20.867938828284609885560800475876, −19.8640085037921595137458476038, −18.68162947032440671204000151203, −17.784442375777429013969157032586, −16.63712941441340774549725645546, −15.48037785621112433024472241933, −14.53945898433841261599062512819, −13.21439459382012531503656866815, −12.34778162653997889864849912341, −11.07086266680593343504505102355, −10.683721730436244363798733360, −9.49028286470654254750411119050, −8.322594921888685890997712019240, −6.12149064624250935149683500293, −5.5229343984319062520579564918, −4.09499836811286741095326783984, −3.26020447188779860934641224811, −1.32809150676557367978701509294, 0.18928703330587124837297749785, 2.2247670193618861857594794353, 4.231793014229701290400790221145, 5.16680750234381032320682298446, 6.46831967549059132920100907044, 7.07323376735794772432924220502, 8.25520076683178398614580723305, 9.55794222333282776643302660620, 11.17662090518755357683339856207, 12.20544027939462577284059457302, 13.114412732062881650357145930953, 13.9902047070260166673328397626, 15.22581949002911180873967768762, 16.278518623519957725252688881633, 17.081264523887049674331650738442, 18.07345810011658611761145741058, 18.70930552367435150010934678406, 20.28189828412627092057542284533, 21.59470487310237245626067070115, 22.501927011988799159800213123035, 23.31933263968854612427049726036, 23.96150945610319457759309959733, 24.990555119498017957371623942735, 25.72703621799631791360295874660, 26.84979428876926058597261515576

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