Properties

Label 1-475-475.177-r1-0-0
Degree $1$
Conductor $475$
Sign $0.791 - 0.611i$
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  + (0.469 − 0.882i)2-s + (−0.970 + 0.241i)3-s + (−0.559 − 0.829i)4-s + (−0.241 + 0.970i)6-s + (0.866 − 0.5i)7-s + (−0.994 + 0.104i)8-s + (0.882 − 0.469i)9-s + (0.669 + 0.743i)11-s + (0.743 + 0.669i)12-s + (0.529 − 0.848i)13-s + (−0.0348 − 0.999i)14-s + (−0.374 + 0.927i)16-s + (−0.898 + 0.438i)17-s i·18-s + (−0.719 + 0.694i)21-s + (0.970 − 0.241i)22-s + ⋯
L(s)  = 1  + (0.469 − 0.882i)2-s + (−0.970 + 0.241i)3-s + (−0.559 − 0.829i)4-s + (−0.241 + 0.970i)6-s + (0.866 − 0.5i)7-s + (−0.994 + 0.104i)8-s + (0.882 − 0.469i)9-s + (0.669 + 0.743i)11-s + (0.743 + 0.669i)12-s + (0.529 − 0.848i)13-s + (−0.0348 − 0.999i)14-s + (−0.374 + 0.927i)16-s + (−0.898 + 0.438i)17-s i·18-s + (−0.719 + 0.694i)21-s + (0.970 − 0.241i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.791 - 0.611i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ 0.791 - 0.611i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.783503181 - 0.6083387002i\)
\(L(\frac12)\) \(\approx\) \(1.783503181 - 0.6083387002i\)
\(L(1)\) \(\approx\) \(1.024203131 - 0.4291866034i\)
\(L(1)\) \(\approx\) \(1.024203131 - 0.4291866034i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.469 - 0.882i)T \)
3 \( 1 + (-0.970 + 0.241i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (0.669 + 0.743i)T \)
13 \( 1 + (0.529 - 0.848i)T \)
17 \( 1 + (-0.898 + 0.438i)T \)
23 \( 1 + (0.788 + 0.615i)T \)
29 \( 1 + (-0.438 + 0.898i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (-0.374 + 0.927i)T \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (0.898 + 0.438i)T \)
53 \( 1 + (0.829 - 0.559i)T \)
59 \( 1 + (-0.990 + 0.139i)T \)
61 \( 1 + (-0.615 + 0.788i)T \)
67 \( 1 + (0.694 - 0.719i)T \)
71 \( 1 + (0.961 + 0.275i)T \)
73 \( 1 + (-0.529 - 0.848i)T \)
79 \( 1 + (0.241 + 0.970i)T \)
83 \( 1 + (0.406 - 0.913i)T \)
89 \( 1 + (0.374 + 0.927i)T \)
97 \( 1 + (-0.694 - 0.719i)T \)
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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

−23.801520033629444354697551781891, −22.91683631857881535515922335429, −22.17476675903240649519649438476, −21.48181331749261522404337901369, −20.720428107727242932561972282362, −18.90404807605751087069976543499, −18.49542041224671251260919917025, −17.30311061280307422497196860071, −17.016853165450304334345598733237, −15.89733621664142824718168992550, −15.27579142813173724822759390951, −14.03038619645457328584891369045, −13.507963315127956997725857166401, −12.23944803503637802014827795497, −11.6374127040188480139550973439, −10.84874112772293266493274786602, −9.16447644235541474298724405851, −8.49007032394530545092970884102, −7.26877886519704063101946788163, −6.46771099231129921716219941911, −5.682597050976764340027201588283, −4.74448619717542881916214109632, −3.92900176393155840048430002775, −2.17856823273596874794700001804, −0.63092897341489255231736058650, 0.93922306015606334671892473930, 1.70421920065142798900043651760, 3.39691975528761862603279810495, 4.4218714201267295365553961894, 5.022794396291688722282317350005, 6.110652107946924133809398588188, 7.13986790256523373521989397175, 8.62372018988865307545590231251, 9.72191666178645805509321218252, 10.67922800333013550228798401604, 11.130131436589200183995012789602, 12.0482539901637379952844409797, 12.86894102629994658294275935770, 13.76324622572815613271685244614, 14.945698059785519963330798872457, 15.46517821879420191450771231543, 16.93992814416972688642043290803, 17.705752593051278853945395402809, 18.17355098189969647587925895097, 19.4809259894484331426925928734, 20.31339264580423370235518779488, 21.04196195335918186130052507433, 21.83169496394254834244181999338, 22.71623633054585558278443512693, 23.19641595457958664263375578703

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