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Properties

Label 1-175-175.2-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: $\chi_{175} (2, \cdot )$
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

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

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