Properties

Label 1-475-475.227-r0-0-0
Degree $1$
Conductor $475$
Sign $-0.728 - 0.684i$
Analytic cond. $2.20589$
Root an. cond. $2.20589$
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  + (−0.951 − 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 + 0.587i)4-s + (−0.809 + 0.587i)6-s + i·7-s + (−0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s + (0.309 − 0.951i)11-s + (0.951 − 0.309i)12-s + (−0.951 + 0.309i)13-s + (0.309 − 0.951i)14-s + (0.309 + 0.951i)16-s + (−0.587 − 0.809i)17-s + i·18-s + (0.809 + 0.587i)21-s + (−0.587 + 0.809i)22-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 + 0.587i)4-s + (−0.809 + 0.587i)6-s + i·7-s + (−0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s + (0.309 − 0.951i)11-s + (0.951 − 0.309i)12-s + (−0.951 + 0.309i)13-s + (0.309 − 0.951i)14-s + (0.309 + 0.951i)16-s + (−0.587 − 0.809i)17-s + i·18-s + (0.809 + 0.587i)21-s + (−0.587 + 0.809i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.728 - 0.684i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ -0.728 - 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2779135500 - 0.7019295020i\)
\(L(\frac12)\) \(\approx\) \(0.2779135500 - 0.7019295020i\)
\(L(1)\) \(\approx\) \(0.6517394925 - 0.3622172112i\)
\(L(1)\) \(\approx\) \(0.6517394925 - 0.3622172112i\)

Euler product

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

−24.38449929436642311178961641789, −23.43543522901276312327733708538, −22.39100115545489649874036540069, −21.430050247783912719230113468582, −20.247899408098180787100146613694, −20.01953482680709813783318079943, −19.30947715275087941467263927360, −17.93403307921264799365636792384, −17.19838890694553526722602816120, −16.55209020369281409278594187128, −15.551786018013792551437028064342, −14.82104470917191577509431122364, −14.16409529154992125915184162806, −12.87914242880683382665942075257, −11.54168274033046468435691931223, −10.52263730861978265897797387973, −9.95693204247832074669997415902, −9.25331266860814512806292655182, −8.06571684253400122359232600286, −7.470597906808018460809330733765, −6.38290549336021527585845142284, −4.967119416410143960171808012737, −4.05352751055973082535976527827, −2.69504906630444064563751633543, −1.56212460132342873123090894021, 0.504691725831594059558312230307, 2.120650402133815512111415691390, 2.55431168208992679430489969813, 3.828860499111560694802664579676, 5.71428048602336935718197664821, 6.64692248948318422602313929253, 7.578543008813114522737416430625, 8.51314556732899202121930821059, 9.10385760666157524462794938229, 9.97146205133424205187256306123, 11.54801146228659463158608898873, 11.83667333467019552689733120700, 12.86413873609224377796304457389, 13.89207778829754007878140948602, 14.94066224697025708753205093328, 15.80620078692389677175979106763, 16.86139429744979240287774052860, 17.752328763451807347614873190, 18.63158245112451719187721813403, 19.02500096339739228093603531456, 19.886481802360747542146165814000, 20.669199649735922764706781561813, 21.67634033984477836622391230194, 22.382051796472326072359319771877, 24.04036003370285709765712294619

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