Properties

Label 1-475-475.106-r0-0-0
Degree $1$
Conductor $475$
Sign $0.345 + 0.938i$
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  + (0.913 + 0.406i)2-s + (−0.978 + 0.207i)3-s + (0.669 + 0.743i)4-s + (−0.978 − 0.207i)6-s + 7-s + (0.309 + 0.951i)8-s + (0.913 − 0.406i)9-s + (−0.809 + 0.587i)11-s + (−0.809 − 0.587i)12-s + (0.913 − 0.406i)13-s + (0.913 + 0.406i)14-s + (−0.104 + 0.994i)16-s + (0.669 − 0.743i)17-s + 18-s + (−0.978 + 0.207i)21-s + (−0.978 + 0.207i)22-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)2-s + (−0.978 + 0.207i)3-s + (0.669 + 0.743i)4-s + (−0.978 − 0.207i)6-s + 7-s + (0.309 + 0.951i)8-s + (0.913 − 0.406i)9-s + (−0.809 + 0.587i)11-s + (−0.809 − 0.587i)12-s + (0.913 − 0.406i)13-s + (0.913 + 0.406i)14-s + (−0.104 + 0.994i)16-s + (0.669 − 0.743i)17-s + 18-s + (−0.978 + 0.207i)21-s + (−0.978 + 0.207i)22-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.345 + 0.938i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ 0.345 + 0.938i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.626270847 + 1.134409254i\)
\(L(\frac12)\) \(\approx\) \(1.626270847 + 1.134409254i\)
\(L(1)\) \(\approx\) \(1.415017666 + 0.5822042990i\)
\(L(1)\) \(\approx\) \(1.415017666 + 0.5822042990i\)

Euler product

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

−23.68175729466134646722406747480, −22.94982338090790233731967421466, −21.86887682662105171078248589228, −21.20277389995747795770222757850, −20.75957971197689853521347812713, −19.26773081739147056395612585274, −18.65700140900776383972097533304, −17.714311328441871637797067124042, −16.69813727920080733566051560433, −15.79009405654809527346307014359, −15.04309676387018262612718980366, −13.7549764880282199758066493473, −13.32326825536086818701200272794, −12.10424959886514097072165206542, −11.515083017445971377508807012317, −10.77101091456171595439795879635, −10.0363994868763861451589526327, −8.32499096609687155159971859740, −7.32378785307276168861343514428, −6.08732409714974534091502703347, −5.53567217289224754648790745406, −4.58390360173608528377379458548, −3.588508278024534776106750686448, −2.042435035818415758620588846988, −1.09650001040304650436159404768, 1.40447969663582334012973243834, 2.8781984954408669632984139077, 4.22268414559324635353904569497, 5.01798006757948999906702277028, 5.62501776069168580327482061302, 6.7692683052234028046418670174, 7.62326839757284346907928888653, 8.6338184516673788818656310044, 10.372597402611416180321406125462, 10.88640353192122183944190140597, 11.967099845071580487758581768121, 12.50140920691583912788814318019, 13.596823947690475520301043823723, 14.51237023952871229139211666612, 15.48818186133621521548470872263, 16.06756788596466314136710735331, 16.98536238992343918752454549106, 17.92998785858090573907112441325, 18.39542375449473660825646257388, 20.233280945308984577201938931025, 20.927662729200425270552994869, 21.44651029177407804842992398334, 22.51056665216238112357844946902, 23.24905414356205095880437578552, 23.64698260076302655553458835568

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