Properties

Label 1-175-175.16-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: $\chi_{175} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (0:\ ),\ -0.968 - 0.249i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1642672822 - 1.297474557i\)
\(L(\frac12)\) \(\approx\) \(0.1642672822 - 1.297474557i\)
\(L(1)\) \(\approx\) \(0.7786967758 - 0.9821243948i\)
\(L(1)\) \(\approx\) \(0.7786967758 - 0.9821243948i\)

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

−27.79967019567007536939760987017, −26.45355974109626171498120813585, −26.16975343886530514070163060992, −25.1105858415240338084214365192, −23.71923827348176848892157923009, −23.23249282561039204990348729514, −22.09850267621163435899115235872, −21.2214313872373286027723593884, −20.73953336188675170829177698110, −19.14701318451448506749097212958, −17.68406651529292797781639347231, −16.856892528526005806532594054141, −15.87650525037131836739259113272, −15.24412407677692047534357506752, −14.1993236356936533245865420434, −13.213160743019847050206547212335, −11.9144822758406977261570556794, −10.892359747839694771869406527502, −9.52977767593473137014490645103, −8.49179678629137549822494292874, −7.24835480422865066717850118856, −5.88291222071180595279237102779, −4.96533861377518445618315854524, −3.94059609882409605610342856913, −2.73838086564835737598505723080, 0.890514188560854994505616415597, 2.37522262837203252425784024436, 3.4036217794543394989831134962, 5.22301518702735139105824050753, 5.94583912293544451065754487388, 7.384467092111278570826658912512, 8.55286469943198705695292073871, 10.17496184096358688084310748947, 11.0398585688973851688288555503, 12.26467570040353935600675066585, 12.869458315422122084675269246496, 13.82317353984285313368922183334, 14.76293226660726868069402136878, 16.05252783242704061930201849259, 17.51724410612537653316532290497, 18.66436918321379726522212930662, 19.00475386793290477092555785403, 20.476894292902590508869084122799, 20.895333425808627357244220111873, 22.51439747155228154602005514344, 23.00831662840763170176618923028, 23.96342931364657485590902947896, 24.80493288511424467535108101118, 25.804479803642809615981399291624, 27.32875989098393703691646654266

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