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Properties

Label 1-175-175.109-r0-0-0
Degree $1$
Conductor $175$
Sign $0.992 + 0.124i$
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  + (0.978 + 0.207i)2-s + (−0.913 − 0.406i)3-s + (0.913 + 0.406i)4-s + (−0.809 − 0.587i)6-s + (0.809 + 0.587i)8-s + (0.669 + 0.743i)9-s + (0.669 − 0.743i)11-s + (−0.669 − 0.743i)12-s + (−0.309 + 0.951i)13-s + (0.669 + 0.743i)16-s + (0.104 − 0.994i)17-s + (0.5 + 0.866i)18-s + (0.913 − 0.406i)19-s + (0.809 − 0.587i)22-s + (0.978 + 0.207i)23-s + (−0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)2-s + (−0.913 − 0.406i)3-s + (0.913 + 0.406i)4-s + (−0.809 − 0.587i)6-s + (0.809 + 0.587i)8-s + (0.669 + 0.743i)9-s + (0.669 − 0.743i)11-s + (−0.669 − 0.743i)12-s + (−0.309 + 0.951i)13-s + (0.669 + 0.743i)16-s + (0.104 − 0.994i)17-s + (0.5 + 0.866i)18-s + (0.913 − 0.406i)19-s + (0.809 − 0.587i)22-s + (0.978 + 0.207i)23-s + (−0.5 − 0.866i)24-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.992 + 0.124i$
Analytic conductor: \(0.812696\)
Root analytic conductor: \(0.812696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (0:\ ),\ 0.992 + 0.124i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.641925979 + 0.1028534000i\)
\(L(\frac12)\) \(\approx\) \(1.641925979 + 0.1028534000i\)
\(L(1)\) \(\approx\) \(1.486003528 + 0.06583206205i\)
\(L(1)\) \(\approx\) \(1.486003528 + 0.06583206205i\)

Euler product

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

−27.81159273454333117725372173838, −26.54508489200792070508186221839, −25.1656002954578352593429188395, −24.364720509797212243728455747817, −23.250607734566948944114899228647, −22.58208009691319210985111635360, −21.92933734202361751474076748851, −20.81124625651084080029648784082, −20.03967984291130138880626637858, −18.71064713890709229612190661933, −17.34448176217936082990476805518, −16.60828517198298398959235594911, −15.26457318890761100723883733449, −14.865063745278816968481971856361, −13.262207535654859462780428481174, −12.372858155754381336930824680213, −11.556923946422013662486353204474, −10.4845057103007778948171973717, −9.63197676030501020562153638679, −7.56876732135832529000932072619, −6.40301891847971339055985226781, −5.4452521546002407711352064947, −4.43747948274973468633458485985, −3.313621493621574898980398114076, −1.47122300603757253991641231215, 1.53211191281850162920610206652, 3.20953828417242849753553125061, 4.67765013937385632418007322965, 5.54928383692710207052147959423, 6.74373587060412769249469931929, 7.39236252157045687353736731548, 9.16973391762479009366382558525, 10.89434533831841411467878343090, 11.5935814205509722452807748942, 12.41407426406071652275923781043, 13.60212533796592259306903070227, 14.31239437783251168530450649602, 15.82794803410969071363103361182, 16.51536424596425258946678409378, 17.39765302736948720059637605684, 18.71804793290753731003475582755, 19.727567802842522922654481158170, 21.07181134589556703709703303167, 21.96220937040105199841613169083, 22.636603490607496813328277053727, 23.6117636640397170261963485193, 24.4166563980227217550966984350, 25.036416321750447293207061822229, 26.457552585445231388401655042532, 27.51935104396675066070414952745

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