Properties

Label 1-175-175.72-r1-0-0
Degree $1$
Conductor $175$
Sign $-0.309 + 0.950i$
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  + (−0.994 − 0.104i)2-s + (−0.207 + 0.978i)3-s + (0.978 + 0.207i)4-s + (0.309 − 0.951i)6-s + (−0.951 − 0.309i)8-s + (−0.913 − 0.406i)9-s + (0.913 − 0.406i)11-s + (−0.406 + 0.913i)12-s + (0.587 + 0.809i)13-s + (0.913 + 0.406i)16-s + (−0.743 + 0.669i)17-s + (0.866 + 0.5i)18-s + (0.978 − 0.207i)19-s + (−0.951 + 0.309i)22-s + (0.994 + 0.104i)23-s + (0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (−0.994 − 0.104i)2-s + (−0.207 + 0.978i)3-s + (0.978 + 0.207i)4-s + (0.309 − 0.951i)6-s + (−0.951 − 0.309i)8-s + (−0.913 − 0.406i)9-s + (0.913 − 0.406i)11-s + (−0.406 + 0.913i)12-s + (0.587 + 0.809i)13-s + (0.913 + 0.406i)16-s + (−0.743 + 0.669i)17-s + (0.866 + 0.5i)18-s + (0.978 − 0.207i)19-s + (−0.951 + 0.309i)22-s + (0.994 + 0.104i)23-s + (0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.309 + 0.950i$
Analytic conductor: \(18.8063\)
Root analytic conductor: \(18.8063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (72, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (1:\ ),\ -0.309 + 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5733012994 + 0.7895325222i\)
\(L(\frac12)\) \(\approx\) \(0.5733012994 + 0.7895325222i\)
\(L(1)\) \(\approx\) \(0.6538402016 + 0.2729548557i\)
\(L(1)\) \(\approx\) \(0.6538402016 + 0.2729548557i\)

Euler product

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

−26.99870631494576727101090265809, −25.82557612114716902230038938638, −24.925967352705712666385585934295, −24.50242485361902292618568636680, −23.19926605172500067603022283552, −22.32176831536412255873897516132, −20.57985082990114169404296325278, −19.962434370828132019657005149252, −18.94156180162392253882177789538, −18.06272002391845105683569133783, −17.41010874190025506219541893517, −16.39166350442334939009288598241, −15.21879079688749626187589439443, −14.01014603927019301954817612033, −12.73660018378259487161593022154, −11.690350626197972579152749172106, −10.88548436698463232434626840592, −9.45029897096348192362579982255, −8.472967644778310111254825486496, −7.34332001000939466304517548146, −6.58889437703317213086768496712, −5.38155953138030833928647791166, −3.1137282136515459332946258416, −1.70667332592244000667296359018, −0.57473692018332749159404291965, 1.22387695487708211814060442558, 3.023380955802214981236883622957, 4.21965844127345439613638756717, 5.88942098009769029414193379424, 6.90095003790407169905038855639, 8.585419352478314368409740010949, 9.14937081077560526840638322084, 10.24910407590878160080513199978, 11.27398157332485182911024633724, 11.891007888765538495332583777612, 13.73648349812622950987462492921, 15.058694343933003726761480735684, 15.86074914727926670226895293514, 16.84381587141367738415031727305, 17.45198316458614379310065438175, 18.776635981397684055931233085812, 19.715154364826232927697356495391, 20.66739718160082551669600870286, 21.52561459792401881746817841261, 22.40018942877772321382133697594, 23.787657121141655755698342436553, 24.870731451950152437115756398902, 25.90940326666516649621457390031, 26.76344750614363758891125194475, 27.28914749016952293952357765310

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