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Properties

Label 1-175-175.88-r1-0-0
Degree $1$
Conductor $175$
Sign $-0.124 + 0.992i$
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  + (−0.207 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.124 + 0.992i$
Analytic conductor: \(18.8063\)
Root analytic conductor: \(18.8063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (1:\ ),\ -0.124 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8757285422 + 0.9927752095i\)
\(L(\frac12)\) \(\approx\) \(0.8757285422 + 0.9927752095i\)
\(L(1)\) \(\approx\) \(0.7098546976 + 0.5284415446i\)
\(L(1)\) \(\approx\) \(0.7098546976 + 0.5284415446i\)

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

−27.50175825584828662136104053230, −25.77495794047680735327172338093, −25.309584763882078820163679548080, −23.68100728300892803703224697464, −23.10045451009064584824651196952, −22.15684949158499376739656896606, −21.03626841790220692992551052017, −19.96084138872188164107042111106, −19.19020882069177817425197855322, −18.24588617570231508172831391902, −17.486084883418445605030965999095, −16.57713559284728149956675677323, −14.78480290680979433432253521074, −13.58951023673571419362194923587, −12.8136287684354252032019952814, −11.866239266569761815064108788917, −11.06496272153164132951886054114, −9.86652779877339921837774814094, −8.60113888197053586866813266505, −7.574093447120648585623059269579, −6.2118268213419774736101482713, −4.83894264766361720044384771167, −3.3654076668055045098935633493, −1.91777061315017027942448609092, −0.86462292271875352080897419007, 0.86185416157747669208217600631, 3.55181273783868916059784288436, 4.52918384400909930228385759307, 5.85342607884694630900003459786, 6.501943495501417439244283912889, 8.24728888264977812643997205790, 9.00939275939865362637266902843, 10.15624795558057495976175812817, 11.11378673651164217301342554008, 12.54099106498230981266615823650, 14.070289920889321211355198750118, 14.68168736611905055630788810997, 15.92892225754684465130574741568, 16.50348485267467359282505901494, 17.33256106781536343458005963915, 18.487861777199374277928176079150, 19.48565391337970508855076116565, 21.01976314486332491423099907916, 21.75218536571527942266298812747, 22.9551931130357187725968892485, 23.41347797792447050381123760581, 24.7083219351963363427154718420, 25.65179870872944761057240638943, 26.53270278498729711468089471700, 27.3680482437769275291088623356

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