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Properties

Label 1-175-175.38-r0-0-0
Degree $1$
Conductor $175$
Sign $0.743 - 0.668i$
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  + (−0.207 + 0.978i)2-s + (0.406 − 0.913i)3-s + (−0.913 − 0.406i)4-s + (0.809 + 0.587i)6-s + (0.587 − 0.809i)8-s + (−0.669 − 0.743i)9-s + (0.669 − 0.743i)11-s + (−0.743 + 0.669i)12-s + (−0.951 − 0.309i)13-s + (0.669 + 0.743i)16-s + (−0.994 − 0.104i)17-s + (0.866 − 0.5i)18-s + (0.913 − 0.406i)19-s + (0.587 + 0.809i)22-s + (0.207 − 0.978i)23-s + (−0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (−0.207 + 0.978i)2-s + (0.406 − 0.913i)3-s + (−0.913 − 0.406i)4-s + (0.809 + 0.587i)6-s + (0.587 − 0.809i)8-s + (−0.669 − 0.743i)9-s + (0.669 − 0.743i)11-s + (−0.743 + 0.669i)12-s + (−0.951 − 0.309i)13-s + (0.669 + 0.743i)16-s + (−0.994 − 0.104i)17-s + (0.866 − 0.5i)18-s + (0.913 − 0.406i)19-s + (0.587 + 0.809i)22-s + (0.207 − 0.978i)23-s + (−0.5 − 0.866i)24-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.743 - 0.668i$
Analytic conductor: \(0.812696\)
Root analytic conductor: \(0.812696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (0:\ ),\ 0.743 - 0.668i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8907965676 - 0.3413895480i\)
\(L(\frac12)\) \(\approx\) \(0.8907965676 - 0.3413895480i\)
\(L(1)\) \(\approx\) \(0.9409911648 - 0.05162205640i\)
\(L(1)\) \(\approx\) \(0.9409911648 - 0.05162205640i\)

Euler product

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

−27.30807576479138540130124939642, −27.04366703037327685810999107529, −25.91667272511550984578635710243, −24.96324212579941258296153254693, −23.384600884166805837206660316, −22.138464609717774977426256468983, −21.9065841774375170533041133101, −20.59638500490490867690079135853, −19.959562788526721319056985454338, −19.20076131762369899680695237006, −17.74074718981185182991015654426, −16.99734218000178761046819468162, −15.700295801666264558616168097685, −14.51977975237304394744823290994, −13.724506226064332020438553307583, −12.3236628505665603191999651859, −11.42091592954115625951604456364, −10.24609161143389013313329153151, −9.50011415531094267605863351818, −8.64958520300246290662355924932, −7.25729749795928430195406786922, −5.15925705061530525015595872566, −4.268643281452746909798810959034, −3.129754860830062054407456861577, −1.86203550601468390685631775609, 0.836868240131738980916632868942, 2.71133601440052498707744450211, 4.40553800958824927597439168329, 5.89370297112763713051070746661, 6.794787650249806139051739118968, 7.76969756240954648680751105091, 8.74597836330231671520241666503, 9.67352086342910806909799615000, 11.368924236059839968451246150717, 12.67334895928513070870698115293, 13.6495341554178564737091997689, 14.425217218567680179130793626213, 15.39199643587575067352583445169, 16.67966959508187382337861710569, 17.55487270947594653613939833660, 18.414438892211159492728645954447, 19.38885113415040953399338708688, 20.1324727312401722253627502691, 21.89992265438200281712737630488, 22.72843326842802933606130670682, 23.87018280440075054279956963260, 24.72226408380044192664815681243, 24.97027797428320196584063374500, 26.548136953226113909527394421497, 26.73854943966024624721945806547

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