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Properties

Label 1-475-475.372-r1-0-0
Degree $1$
Conductor $475$
Sign $0.712 + 0.702i$
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  + (−0.994 − 0.104i)2-s + (−0.743 − 0.669i)3-s + (0.978 + 0.207i)4-s + (0.669 + 0.743i)6-s + i·7-s + (−0.951 − 0.309i)8-s + (0.104 + 0.994i)9-s + (−0.809 − 0.587i)11-s + (−0.587 − 0.809i)12-s + (−0.994 + 0.104i)13-s + (0.104 − 0.994i)14-s + (0.913 + 0.406i)16-s + (−0.207 − 0.978i)17-s − i·18-s + (0.669 − 0.743i)21-s + (0.743 + 0.669i)22-s + ⋯
L(s)  = 1  + (−0.994 − 0.104i)2-s + (−0.743 − 0.669i)3-s + (0.978 + 0.207i)4-s + (0.669 + 0.743i)6-s + i·7-s + (−0.951 − 0.309i)8-s + (0.104 + 0.994i)9-s + (−0.809 − 0.587i)11-s + (−0.587 − 0.809i)12-s + (−0.994 + 0.104i)13-s + (0.104 − 0.994i)14-s + (0.913 + 0.406i)16-s + (−0.207 − 0.978i)17-s − i·18-s + (0.669 − 0.743i)21-s + (0.743 + 0.669i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.712 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.712 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.712 + 0.702i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (372, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ 0.712 + 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3642637446 + 0.1493790799i\)
\(L(\frac12)\) \(\approx\) \(0.3642637446 + 0.1493790799i\)
\(L(1)\) \(\approx\) \(0.4503122221 - 0.07561597076i\)
\(L(1)\) \(\approx\) \(0.4503122221 - 0.07561597076i\)

Euler product

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

−23.66524088304814614611767195492, −22.763485894364738780189264284945, −21.59689986249433195092333864601, −20.91247875783001263185367629139, −19.999213969129587958567327027287, −19.338163742629273410829965646944, −17.97334699305112174009182713524, −17.45065737128251892571466727002, −16.880872308746775567529449093179, −15.85479024186115584701516981450, −15.294283733823957171421053813305, −14.233605362945589465309908533852, −12.73629720222728291013064627982, −11.8884056427976514711994975786, −10.822424195524983057527269247561, −10.20372883094569510925847025835, −9.69948302610517910035481499596, −8.358041527441592093522855181338, −7.366348011449221118223746077, −6.575853980927132423692745569060, −5.408499190509144966679865249416, −4.40219248952937194986184446606, −3.08594415260527440347084816563, −1.57409145976809325606900529584, −0.2510618622319499863686025449, 0.65544458427810985657354562829, 2.19517102598325799467790844342, 2.7515911407969182230095026476, 4.89771316142102350953439368853, 5.84186454957076318716639346855, 6.72955943930217398085445998500, 7.712762620871432980497588902202, 8.47763903971650227785434892364, 9.58344349738425632151518432573, 10.54268599582956452309773736259, 11.467874313250183356358531367522, 12.132946101845859399481302255352, 12.87567642881187475177618654296, 14.19849515298857029722653501918, 15.488149021636647106656223358310, 16.178585254670962982349972271743, 16.97797989244041605850598792256, 17.99767081053446628216652514083, 18.437210374316791648554066247723, 19.149342737843647429119971206194, 20.02620674513294219034986465675, 21.23623590117663693074271663126, 21.893218539862355726026287818331, 22.87528261944063971383093262960, 24.07304609831491722261584844046

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