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Properties

Label 1-475-475.154-r1-0-0
Degree $1$
Conductor $475$
Sign $-0.856 - 0.516i$
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  + (0.559 + 0.829i)2-s + (−0.882 + 0.469i)3-s + (−0.374 + 0.927i)4-s + (−0.882 − 0.469i)6-s + (0.5 − 0.866i)7-s + (−0.978 + 0.207i)8-s + (0.559 − 0.829i)9-s + (−0.104 + 0.994i)11-s + (−0.104 − 0.994i)12-s + (0.438 + 0.898i)13-s + (0.997 − 0.0697i)14-s + (−0.719 − 0.694i)16-s + (0.615 − 0.788i)17-s + 18-s + (−0.0348 + 0.999i)21-s + (−0.882 + 0.469i)22-s + ⋯
L(s)  = 1  + (0.559 + 0.829i)2-s + (−0.882 + 0.469i)3-s + (−0.374 + 0.927i)4-s + (−0.882 − 0.469i)6-s + (0.5 − 0.866i)7-s + (−0.978 + 0.207i)8-s + (0.559 − 0.829i)9-s + (−0.104 + 0.994i)11-s + (−0.104 − 0.994i)12-s + (0.438 + 0.898i)13-s + (0.997 − 0.0697i)14-s + (−0.719 − 0.694i)16-s + (0.615 − 0.788i)17-s + 18-s + (−0.0348 + 0.999i)21-s + (−0.882 + 0.469i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.856 - 0.516i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ -0.856 - 0.516i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3259714600 + 1.172529658i\)
\(L(\frac12)\) \(\approx\) \(-0.3259714600 + 1.172529658i\)
\(L(1)\) \(\approx\) \(0.7241849892 + 0.6921846515i\)
\(L(1)\) \(\approx\) \(0.7241849892 + 0.6921846515i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.559 + 0.829i)T \)
3 \( 1 + (-0.882 + 0.469i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (0.438 + 0.898i)T \)
17 \( 1 + (0.615 - 0.788i)T \)
23 \( 1 + (0.241 + 0.970i)T \)
29 \( 1 + (0.615 + 0.788i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (0.719 + 0.694i)T \)
43 \( 1 + (-0.766 + 0.642i)T \)
47 \( 1 + (0.615 + 0.788i)T \)
53 \( 1 + (-0.374 + 0.927i)T \)
59 \( 1 + (-0.961 + 0.275i)T \)
61 \( 1 + (-0.241 - 0.970i)T \)
67 \( 1 + (0.0348 + 0.999i)T \)
71 \( 1 + (-0.848 - 0.529i)T \)
73 \( 1 + (-0.438 + 0.898i)T \)
79 \( 1 + (0.882 - 0.469i)T \)
83 \( 1 + (-0.669 - 0.743i)T \)
89 \( 1 + (0.719 - 0.694i)T \)
97 \( 1 + (0.0348 - 0.999i)T \)
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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

−23.00149059278872514107100317973, −22.23312630862693252351727060904, −21.4797813898474677216555186073, −20.88042390202097101829677108578, −19.56542468977311424357125977990, −18.817182980767968506482089507455, −18.224837849582887318671622355843, −17.370305499512947575898251415316, −16.14353175736155180833589862870, −15.250575465341811073314653555984, −14.22401265330190333368424410343, −13.279079500376727579347801310560, −12.42883344083970393155971159700, −11.88428963728440903943715566926, −10.79480866780833400547310573266, −10.462090421608640794232562329326, −8.88698105086500042466811376989, −8.0315935809016680542507632702, −6.426177096459203197477152531949, −5.660182872743500185312696082613, −5.0847238123668499177274163954, −3.70889892121182293035691620763, −2.512425095965668969293577864845, −1.39186443548889267173167299476, −0.31710582501306187131350988497, 1.35310290283735480844934525393, 3.3370449912296622686394190983, 4.41590568297643846347774537311, 4.88906677421817079761282029982, 6.01294154942463745416765388990, 7.03836078752665906279539219901, 7.596853461926323411958423055273, 9.08287443192237831447802325053, 9.947578638188046908203755491480, 11.16512423681375526001803844534, 11.876435817839716902944614739176, 12.840642322946636173249675378209, 13.87298450514310446497070074092, 14.65517728846877154723689824423, 15.607401711874861133176267564628, 16.37733012780256060996377750892, 17.083987259982125364535271821487, 17.772240854186418599936087015617, 18.573090397626623593792488782834, 20.26689550079099898716676017419, 20.96957783185059687310907185233, 21.69696512097158316901683621255, 22.65879674752638267622655025030, 23.50123144036821934404184110053, 23.59830991143583619937068022773

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