Properties

Label 1-175-175.39-r0-0-0
Degree $1$
Conductor $175$
Sign $-0.968 + 0.249i$
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  + (−0.669 − 0.743i)2-s + (0.104 − 0.994i)3-s + (−0.104 + 0.994i)4-s + (−0.809 + 0.587i)6-s + (0.809 − 0.587i)8-s + (−0.978 − 0.207i)9-s + (−0.978 + 0.207i)11-s + (0.978 + 0.207i)12-s + (−0.309 − 0.951i)13-s + (−0.978 − 0.207i)16-s + (−0.913 − 0.406i)17-s + (0.5 + 0.866i)18-s + (−0.104 − 0.994i)19-s + (0.809 + 0.587i)22-s + (−0.669 − 0.743i)23-s + (−0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (−0.669 − 0.743i)2-s + (0.104 − 0.994i)3-s + (−0.104 + 0.994i)4-s + (−0.809 + 0.587i)6-s + (0.809 − 0.587i)8-s + (−0.978 − 0.207i)9-s + (−0.978 + 0.207i)11-s + (0.978 + 0.207i)12-s + (−0.309 − 0.951i)13-s + (−0.978 − 0.207i)16-s + (−0.913 − 0.406i)17-s + (0.5 + 0.866i)18-s + (−0.104 − 0.994i)19-s + (0.809 + 0.587i)22-s + (−0.669 − 0.743i)23-s + (−0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.968 + 0.249i$
Analytic conductor: \(0.812696\)
Root analytic conductor: \(0.812696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (0:\ ),\ -0.968 + 0.249i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05912464177 - 0.4669993766i\)
\(L(\frac12)\) \(\approx\) \(-0.05912464177 - 0.4669993766i\)
\(L(1)\) \(\approx\) \(0.4239781971 - 0.4330713274i\)
\(L(1)\) \(\approx\) \(0.4239781971 - 0.4330713274i\)

Euler product

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

−27.814020231648725958183380229808, −26.69107259828929651458059739935, −26.330052527054533415833874882609, −25.36792239066265822423800158627, −24.17800827675847013278935408284, −23.33883976778571039817444386417, −22.20457175417409079664020628673, −21.176589511740499925246276044821, −20.12739995145778706230380122597, −19.144363374741623743293802795841, −18.06487593592610479393568405882, −16.95054513113265396292027048684, −16.201265805324462030131259601034, −15.36118184151160105243611567129, −14.470242463255042214426023775478, −13.434663850861266095935557788836, −11.543122881821613801618854954273, −10.52177576996073178002524594982, −9.678392874648017623821835006616, −8.67760237056671287845793710893, −7.70235422699559014109851452642, −6.222394618992766235139098394748, −5.18353201464658707534360567208, −4.02161672439618875848739685891, −2.15891177536172109607440030960, 0.43872940117787875001333213541, 2.1703178679136334918387635186, 2.96806069751656257925410395166, 4.8220436451510530022924536420, 6.53797378841781765480681660417, 7.699410557306215871181572250855, 8.400740603107979674424338462850, 9.70348564889078920397797220360, 10.883193677100364857046953324353, 11.83497876170215119967850263633, 12.98613893936222013047560870162, 13.40034230342831368233826992767, 15.069946992869461261958582310584, 16.38262525347989835079526354062, 17.82022633283989060197071683215, 17.93600696237593234414288668258, 19.16927138609531215653255714373, 20.01129426300333793673451998247, 20.7223504623699400916238720618, 22.089615587180995631498393109130, 22.9710167990890110976311930986, 24.21762305220172439635651769751, 25.118950358764121276460329304519, 26.09728311060925853129067454080, 26.83424699399941610471310339103

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