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Properties

Label 1-475-475.211-r1-0-0
Degree $1$
Conductor $475$
Sign $0.667 - 0.744i$
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  + (0.615 − 0.788i)2-s + (−0.438 + 0.898i)3-s + (−0.241 − 0.970i)4-s + (0.438 + 0.898i)6-s + (−0.5 + 0.866i)7-s + (−0.913 − 0.406i)8-s + (−0.615 − 0.788i)9-s + (−0.978 + 0.207i)11-s + (0.978 + 0.207i)12-s + (−0.990 + 0.139i)13-s + (0.374 + 0.927i)14-s + (−0.882 + 0.469i)16-s + (0.961 − 0.275i)17-s − 18-s + (−0.559 − 0.829i)21-s + (−0.438 + 0.898i)22-s + ⋯
L(s)  = 1  + (0.615 − 0.788i)2-s + (−0.438 + 0.898i)3-s + (−0.241 − 0.970i)4-s + (0.438 + 0.898i)6-s + (−0.5 + 0.866i)7-s + (−0.913 − 0.406i)8-s + (−0.615 − 0.788i)9-s + (−0.978 + 0.207i)11-s + (0.978 + 0.207i)12-s + (−0.990 + 0.139i)13-s + (0.374 + 0.927i)14-s + (−0.882 + 0.469i)16-s + (0.961 − 0.275i)17-s − 18-s + (−0.559 − 0.829i)21-s + (−0.438 + 0.898i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.667 - 0.744i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ 0.667 - 0.744i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.291467190 - 0.5767194447i\)
\(L(\frac12)\) \(\approx\) \(1.291467190 - 0.5767194447i\)
\(L(1)\) \(\approx\) \(0.9801201734 - 0.1860986723i\)
\(L(1)\) \(\approx\) \(0.9801201734 - 0.1860986723i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.615 - 0.788i)T \)
3 \( 1 + (-0.438 + 0.898i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.978 + 0.207i)T \)
13 \( 1 + (-0.990 + 0.139i)T \)
17 \( 1 + (0.961 - 0.275i)T \)
23 \( 1 + (0.848 - 0.529i)T \)
29 \( 1 + (-0.961 - 0.275i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (0.882 - 0.469i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (0.961 + 0.275i)T \)
53 \( 1 + (0.241 + 0.970i)T \)
59 \( 1 + (-0.0348 + 0.999i)T \)
61 \( 1 + (0.848 - 0.529i)T \)
67 \( 1 + (-0.559 + 0.829i)T \)
71 \( 1 + (0.997 - 0.0697i)T \)
73 \( 1 + (0.990 + 0.139i)T \)
79 \( 1 + (-0.438 + 0.898i)T \)
83 \( 1 + (-0.104 - 0.994i)T \)
89 \( 1 + (0.882 + 0.469i)T \)
97 \( 1 + (-0.559 - 0.829i)T \)
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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

−23.86462830433526598519943285976, −22.96247255154504048355318529446, −22.48423373752424832961400353336, −21.37254089187505732549654524542, −20.41537010039198591904934773929, −19.28426017179082542813414433015, −18.51089059053291853327377259591, −17.39153414578829828282642623281, −16.91125679974410796247399059711, −16.14107862266718473497070888744, −14.97773204346014033989308385086, −14.08259005066656755802946771204, −13.140250876236029860726419960704, −12.80534410570370315147645403839, −11.74157892456982853888046082391, −10.68599132025482316786857540556, −9.47879808721373473999803134429, −7.91918966526216514869892153800, −7.57058866530278828410897210679, −6.66197826480194175084789766565, −5.65414804739493235932460077395, −4.912659456882229874514252119, −3.51486750056416091148366422796, −2.47539741223314358027838129207, −0.6818185675388889141563391923, 0.492263724815230567131793232066, 2.38077077229191624482171748083, 3.09686958040013177622200612439, 4.281378818304527099570442340898, 5.337792758242123764961999448729, 5.692373557378627906826045456664, 7.1429903389849175381283680846, 8.85575425316246958898128242463, 9.59452066145966700408527099647, 10.358385508863131568207738039534, 11.166286018908796543978244907654, 12.359957945292707307721290434424, 12.528316993633328866972976371801, 14.02690277170136371946710439025, 14.89919199322697464130795225880, 15.549328114149180695501902053205, 16.401541911864655338433332871767, 17.57945213776074157688694001009, 18.59751533883901372487606781548, 19.29377513243201029293308363770, 20.42012866229123113371562623096, 21.136766325137126681968024654167, 21.7010659706393537675003897343, 22.6044941749634802563658147270, 23.05781664832811817860710257705

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