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Properties

Label 1-175-175.81-r0-0-0
Degree $1$
Conductor $175$
Sign $-0.309 - 0.950i$
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  + (−0.104 − 0.994i)2-s + (−0.978 + 0.207i)3-s + (−0.978 + 0.207i)4-s + (0.309 + 0.951i)6-s + (0.309 + 0.951i)8-s + (0.913 − 0.406i)9-s + (0.913 + 0.406i)11-s + (0.913 − 0.406i)12-s + (−0.809 − 0.587i)13-s + (0.913 − 0.406i)16-s + (0.669 − 0.743i)17-s + (−0.5 − 0.866i)18-s + (−0.978 − 0.207i)19-s + (0.309 − 0.951i)22-s + (−0.104 − 0.994i)23-s + (−0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (−0.978 + 0.207i)3-s + (−0.978 + 0.207i)4-s + (0.309 + 0.951i)6-s + (0.309 + 0.951i)8-s + (0.913 − 0.406i)9-s + (0.913 + 0.406i)11-s + (0.913 − 0.406i)12-s + (−0.809 − 0.587i)13-s + (0.913 − 0.406i)16-s + (0.669 − 0.743i)17-s + (−0.5 − 0.866i)18-s + (−0.978 − 0.207i)19-s + (0.309 − 0.951i)22-s + (−0.104 − 0.994i)23-s + (−0.5 − 0.866i)24-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.309 - 0.950i$
Analytic conductor: \(0.812696\)
Root analytic conductor: \(0.812696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (0:\ ),\ -0.309 - 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3864286343 - 0.5321773640i\)
\(L(\frac12)\) \(\approx\) \(0.3864286343 - 0.5321773640i\)
\(L(1)\) \(\approx\) \(0.5991498146 - 0.3538206768i\)
\(L(1)\) \(\approx\) \(0.5991498146 - 0.3538206768i\)

Euler product

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

−27.52103761508917609131021592608, −26.96323151371822695863770395881, −25.62010558562656556955551855660, −24.77514312509440883968559075290, −23.77685318420403458979734777685, −23.29751378196187265679604645164, −21.986551358336729187840307020541, −21.61683161659793623754322939529, −19.47112505285521967688371639955, −18.8351048930787632775851023758, −17.561346310918603550669635299511, −16.9758713737020358761235640470, −16.2217580062626356151067776861, −14.98593262587612608081050310940, −14.03451216559986506758748339674, −12.78902041947419161145455760275, −11.8564472059763379279893453725, −10.48657982645686826505647486929, −9.42278760697104008294197690664, −8.11627201273913744953874477590, −6.90988745922300722920845815673, −6.166117138267297439879512712748, −5.04594904218904845948872270322, −3.92661373203095221302136872646, −1.34849712208955929165871844636, 0.731175383048614029533246381153, 2.41210006249961863193273653501, 4.06337119122809419283552431195, 4.91334405684773618675941733421, 6.26741943204918755238204100585, 7.74173938162390803466398436433, 9.317465709158754342350929277610, 10.09795006109800551014182901073, 11.09789015888049958272324929078, 12.1131743449700737022215571074, 12.66436239259256386842808214713, 14.11956739831658012286449491308, 15.274060195666681432694302581775, 16.85572508854984183327448209301, 17.32821536172350489314798747368, 18.42196690459418614440155235519, 19.364380009138838494503645502074, 20.4862794085152500167976098429, 21.38170061812475340606411630649, 22.46180242452760720163093439223, 22.78595981028171220114665069714, 24.01870436678240683640179070527, 25.25781365944731858440901299621, 26.719297393321046102762162161332, 27.43131199834350277191286925313

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