Properties

Label 1-475-475.178-r1-0-0
Degree $1$
Conductor $475$
Sign $0.938 + 0.345i$
Analytic cond. $51.0458$
Root an. cond. $51.0458$
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  + (−0.406 − 0.913i)2-s + (0.207 − 0.978i)3-s + (−0.669 + 0.743i)4-s + (−0.978 + 0.207i)6-s i·7-s + (0.951 + 0.309i)8-s + (−0.913 − 0.406i)9-s + (−0.809 − 0.587i)11-s + (0.587 + 0.809i)12-s + (−0.406 + 0.913i)13-s + (−0.913 + 0.406i)14-s + (−0.104 − 0.994i)16-s + (0.743 − 0.669i)17-s + i·18-s + (−0.978 − 0.207i)21-s + (−0.207 + 0.978i)22-s + ⋯
L(s)  = 1  + (−0.406 − 0.913i)2-s + (0.207 − 0.978i)3-s + (−0.669 + 0.743i)4-s + (−0.978 + 0.207i)6-s i·7-s + (0.951 + 0.309i)8-s + (−0.913 − 0.406i)9-s + (−0.809 − 0.587i)11-s + (0.587 + 0.809i)12-s + (−0.406 + 0.913i)13-s + (−0.913 + 0.406i)14-s + (−0.104 − 0.994i)16-s + (0.743 − 0.669i)17-s + i·18-s + (−0.978 − 0.207i)21-s + (−0.207 + 0.978i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.938 + 0.345i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ 0.938 + 0.345i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1026018048 + 0.01828023723i\)
\(L(\frac12)\) \(\approx\) \(0.1026018048 + 0.01828023723i\)
\(L(1)\) \(\approx\) \(0.4559936116 - 0.4949628877i\)
\(L(1)\) \(\approx\) \(0.4559936116 - 0.4949628877i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.406 - 0.913i)T \)
3 \( 1 + (0.207 - 0.978i)T \)
7 \( 1 - iT \)
11 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (-0.406 + 0.913i)T \)
17 \( 1 + (0.743 - 0.669i)T \)
23 \( 1 + (-0.994 - 0.104i)T \)
29 \( 1 + (-0.669 + 0.743i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (-0.743 - 0.669i)T \)
53 \( 1 + (0.743 + 0.669i)T \)
59 \( 1 + (0.104 + 0.994i)T \)
61 \( 1 + (-0.104 + 0.994i)T \)
67 \( 1 + (0.207 + 0.978i)T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (0.406 + 0.913i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (0.951 + 0.309i)T \)
89 \( 1 + (0.104 - 0.994i)T \)
97 \( 1 + (-0.207 + 0.978i)T \)
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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

−23.51413976271641328246927750852, −22.77816093693307833033357044775, −21.94610137577617203854709121139, −21.15945976883932029102869836623, −20.055192679670888113736585204731, −19.26575400906317013252427367618, −18.16272820785069999904521153417, −17.55951916208614647657835161107, −16.45007278332825744172145690367, −15.77970177038605855773432439088, −15.00267161382753930090181117762, −14.612552784167804859858915891559, −13.30110190917439930434676137317, −12.27481955933586924013438100318, −10.891787298760881090720244848293, −9.95447059261617388443745782968, −9.49527661979827397881238335938, −8.18103402987069810919885388776, −7.902403505515921903979345058057, −6.21489168753279481291744945446, −5.41038796964691229989667692429, −4.73538044715664105183259627087, −3.36488635548971751140035906509, −2.063700504688138190569100876499, −0.0355971083453007943270011208, 0.94877386781738957588642564422, 2.06301242247862486429823911666, 3.07658077766284342881356615102, 4.1203685582564308095130496073, 5.46395879922900811287179216333, 6.952082954662688043137721858234, 7.667391756194623011142768874567, 8.46983352182965359335934705347, 9.59188653192514950554371724372, 10.48372453440403717104338015044, 11.5261723347729562508702796023, 12.1328059950424631449358485159, 13.34284276782804271999703221490, 13.6553942480000027002481219101, 14.59499130004884577137674203006, 16.42773502275560692452888228363, 16.89310349387705430892097158288, 18.01688316525939977800241809030, 18.63670588065451675496141771278, 19.3259035802125751256000119983, 20.24031811099449047331712358137, 20.76141232965637852152431669438, 21.839954159584895873544198891425, 22.85384748107965678462733817945, 23.67244834279610954176480754047

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