Properties

Label 1-175-175.79-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: $\chi_{175} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (0:\ ),\ 0.309 - 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.245929270 - 0.9047035424i\)
\(L(\frac12)\) \(\approx\) \(1.245929270 - 0.9047035424i\)
\(L(1)\) \(\approx\) \(1.244373445 - 0.6173716568i\)
\(L(1)\) \(\approx\) \(1.244373445 - 0.6173716568i\)

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

−27.36961525702346843170902601377, −26.28859610903040491477126558877, −25.71876018368626385063544484674, −24.85755241165538881462649596311, −24.0118684486917789501940591754, −23.11869442604195926618890196109, −21.865247816406610597264467556057, −21.00945617900975459565494713004, −19.618494299311783244974828709483, −18.91775135637475674346648879202, −17.73388697099076326363213229353, −16.8357769227722271941300842651, −15.450066980450256235240362218003, −14.98524738535652114359431587863, −13.773519877439955037177126430602, −13.22252658326908735397103567969, −11.88567387280759298845862942385, −10.04558119714133108116754087221, −8.94546434309461550533334033047, −8.317537521917977996016280734056, −6.99861556630840891605823108704, −6.2550960181692936103280631563, −4.43745176602604246516477279278, −3.6328491882607928321730224395, −1.73607757896762939303955112423, 1.41691159559981761751032487891, 2.79041811243786407248669885394, 3.76010102143533771476447429571, 4.84401508109833114283258721189, 6.59627806505841777797500189879, 8.419887469872464644490548090967, 8.86856129850319245503997527480, 10.14108359997346215795566103563, 11.01696708581199735466598142894, 12.33470134246102297632666800069, 13.3522770286168066680825494168, 14.15345172652931860793332729139, 15.07884994771223544472409601370, 16.39405799034525591568081502112, 17.84347574362127097337196170748, 18.79377457332074520750792070810, 19.68182224871867782280102698128, 20.409422369113264884556750368882, 21.271188849012286728226637301717, 22.16731406093359324259177638021, 23.15550286745664123608081249139, 24.477924819551278348267972124415, 25.40613969701044379148677922429, 26.560480968482734575985896801421, 27.3014631143640481526108347625

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