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Properties

Label 1-775-775.166-r1-0-0
Degree $1$
Conductor $775$
Sign $0.929 + 0.369i$
Analytic cond. $83.2853$
Root an. cond. $83.2853$
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  + (−0.809 − 0.587i)2-s + (0.5 + 0.866i)3-s + (0.309 + 0.951i)4-s + (0.104 − 0.994i)6-s + (−0.978 + 0.207i)7-s + (0.309 − 0.951i)8-s + (−0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s + (−0.669 + 0.743i)12-s + (0.104 + 0.994i)13-s + (0.913 + 0.406i)14-s + (−0.809 + 0.587i)16-s + (0.978 − 0.207i)17-s + (0.913 − 0.406i)18-s + (−0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.5 + 0.866i)3-s + (0.309 + 0.951i)4-s + (0.104 − 0.994i)6-s + (−0.978 + 0.207i)7-s + (0.309 − 0.951i)8-s + (−0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s + (−0.669 + 0.743i)12-s + (0.104 + 0.994i)13-s + (0.913 + 0.406i)14-s + (−0.809 + 0.587i)16-s + (0.978 − 0.207i)17-s + (0.913 − 0.406i)18-s + (−0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $0.929 + 0.369i$
Analytic conductor: \(83.2853\)
Root analytic conductor: \(83.2853\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (166, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 775,\ (1:\ ),\ 0.929 + 0.369i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.377603579 + 0.2634793120i\)
\(L(\frac12)\) \(\approx\) \(1.377603579 + 0.2634793120i\)
\(L(1)\) \(\approx\) \(0.8293248060 + 0.06978857326i\)
\(L(1)\) \(\approx\) \(0.8293248060 + 0.06978857326i\)

Euler product

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

−22.52288838382348997215757586549, −20.891893782217670024459534054392, −20.16649997852327436748472398508, −19.51787570569753503826479667338, −18.863605922707816286110672493184, −18.18622712629013426725895108607, −17.16217320877873519225890347995, −16.79660763489125967384258041629, −15.45745609994149155829887688131, −14.95927258280716596549728531410, −14.07794623690910552635844610359, −13.07711895107232494745605899850, −12.4184828633958098313798067981, −11.33826148019933626660961218774, −9.9512709410183052933567460738, −9.7220496355180621695289408595, −8.51544757054746362928761936581, −7.8030400024174458706979028360, −7.034832426537903245476489186, −6.2793007551386240409581538414, −5.468040567476450440808557503308, −3.80366728588819991753222821252, −2.72803923298123823092587848788, −1.55850401083118069004598503751, −0.64590155364111632726784909106, 0.65086990375028855198041215828, 2.12869479430634718570417232866, 3.16056283727825386529496295660, 3.65953906802554589109799100690, 4.83893293166560391687965506041, 6.257859957908363750096593878560, 7.15282724518756100039889414777, 8.40194374874669006512069880403, 9.07480532408823760319453090411, 9.52528411583511069694242047771, 10.536044264214724006371074165859, 11.208706720318972843364665930108, 12.13455896945232379047332664946, 13.1584398297705622602420777023, 13.979766908676603567456427905233, 15.0365538519370871084550737573, 16.01520085411793463897032489984, 16.61857790838376759175140588929, 17.0410518300307757851710320151, 18.673947093230458565058605403057, 18.9522237993721598908757126622, 19.789604050825643163397749642829, 20.45775193917458758616713190979, 21.53950220919844290738517629678, 21.73671830568617874853324849965

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