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Properties

Label 1-775-775.103-r1-0-0
Degree $1$
Conductor $775$
Sign $-0.844 - 0.535i$
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  + (−0.951 − 0.309i)2-s + (0.866 − 0.5i)3-s + (0.809 + 0.587i)4-s + (−0.978 + 0.207i)6-s + (0.406 − 0.913i)7-s + (−0.587 − 0.809i)8-s + (0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.994 + 0.104i)12-s + (0.207 − 0.978i)13-s + (−0.669 + 0.743i)14-s + (0.309 + 0.951i)16-s + (0.406 − 0.913i)17-s + (−0.743 + 0.669i)18-s + (0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (0.866 − 0.5i)3-s + (0.809 + 0.587i)4-s + (−0.978 + 0.207i)6-s + (0.406 − 0.913i)7-s + (−0.587 − 0.809i)8-s + (0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.994 + 0.104i)12-s + (0.207 − 0.978i)13-s + (−0.669 + 0.743i)14-s + (0.309 + 0.951i)16-s + (0.406 − 0.913i)17-s + (−0.743 + 0.669i)18-s + (0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $-0.844 - 0.535i$
Analytic conductor: \(83.2853\)
Root analytic conductor: \(83.2853\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 775,\ (1:\ ),\ -0.844 - 0.535i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4771158285 - 1.644434482i\)
\(L(\frac12)\) \(\approx\) \(0.4771158285 - 1.644434482i\)
\(L(1)\) \(\approx\) \(0.8630645012 - 0.5093648314i\)
\(L(1)\) \(\approx\) \(0.8630645012 - 0.5093648314i\)

Euler product

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

−22.12065736916381936208259731062, −21.436651970557327640014045345008, −20.8908036445044592322518740079, −19.89053992958580125220061273462, −19.15238402475293090844490329417, −18.610257836461207002960351671710, −17.791454792573955733885856214725, −16.646942819009005870429506403470, −15.93430130290957730197800183599, −15.43030837575999996166083280364, −14.42497046054119817751383002447, −13.93137426660610217111584956860, −12.562629565689670152784107077188, −11.43498465375012215167361845474, −10.76948217857600015937999077196, −9.76655994497220231691319051276, −9.01645119517334978969703135129, −8.36165609609560593595458974153, −7.78252162483153223780283056939, −6.50195560084364733391358114603, −5.59116394661103268728913128894, −4.53185181890159317796668831301, −3.11481225933758506173393785423, −2.321627026484196531645053640937, −1.296369224428213495579923677441, 0.4781847540599271345059414362, 1.38122935454102017916461353981, 2.39561795954652526242315210243, 3.31473778624769387831126854940, 4.306156281894725217099200913977, 5.89069170673596977761460621386, 7.151950296698994996948717218540, 7.73271153115749029273373123876, 8.131839384083687793971174507153, 9.45560282557521333735109815085, 9.99675066164707596148240384518, 10.82365412074612363651177847858, 12.02529018594250462732616886284, 12.63489782454621917405850076892, 13.64450682307596734562942031121, 14.42082136645211937555059662204, 15.47751327704272145364782179440, 16.137396076335194896086411129570, 17.32415513826455918425531123197, 18.01904334656858777807520321514, 18.42356899712977317219064017415, 19.60761260024734370361522340387, 20.13337064684121673467331686071, 20.68902345480850703150492254136, 21.287831716579978530234208688116

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