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Properties

Label 1-775-775.104-r1-0-0
Degree $1$
Conductor $775$
Sign $0.991 + 0.126i$
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  − 2-s + (−0.978 + 0.207i)3-s + 4-s + (0.978 − 0.207i)6-s + (0.978 − 0.207i)7-s − 8-s + (0.913 − 0.406i)9-s + (0.104 + 0.994i)11-s + (−0.978 + 0.207i)12-s + (−0.5 + 0.866i)13-s + (−0.978 + 0.207i)14-s + 16-s + (−0.5 − 0.866i)17-s + (−0.913 + 0.406i)18-s + (0.669 − 0.743i)19-s + ⋯
L(s)  = 1  − 2-s + (−0.978 + 0.207i)3-s + 4-s + (0.978 − 0.207i)6-s + (0.978 − 0.207i)7-s − 8-s + (0.913 − 0.406i)9-s + (0.104 + 0.994i)11-s + (−0.978 + 0.207i)12-s + (−0.5 + 0.866i)13-s + (−0.978 + 0.207i)14-s + 16-s + (−0.5 − 0.866i)17-s + (−0.913 + 0.406i)18-s + (0.669 − 0.743i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(83.2853\)
Root analytic conductor: \(83.2853\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 775,\ (1:\ ),\ 0.991 + 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.043313515 + 0.06614397753i\)
\(L(\frac12)\) \(\approx\) \(1.043313515 + 0.06614397753i\)
\(L(1)\) \(\approx\) \(0.6317338551 + 0.04300968643i\)
\(L(1)\) \(\approx\) \(0.6317338551 + 0.04300968643i\)

Euler product

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

−21.84449370728179083504126197213, −21.46288802829536624665475076418, −20.462718756005712160645283943504, −19.46180085883268582484984645568, −18.78930311477433436905648651586, −17.88748799934133266892713784362, −17.47457133445197848568244920983, −16.75843690852364069477508913726, −15.82652578330120508678766897114, −15.167559828471768318163146042948, −14.04007654146627758962299730095, −12.800650678750323186881703426083, −11.9482684061770309233406522365, −11.262136323098242005997188542276, −10.64974059930950431907875720144, −9.81503908806922840806303025497, −8.6174669072224297665842704551, −7.8743173811485871442851182996, −7.144867286271749153418860108500, −5.825033963275023457857280804821, −5.57960339207894345215731042724, −4.082900144301242912466845521792, −2.63986946049586039320211394947, −1.46444405292489207317473809687, −0.68341873921955398696533976424, 0.629346245834636112028343391650, 1.59161843042376178749402161494, 2.67627059712570743932873887702, 4.501823827240954357374135532274, 4.91266096921028624811677451718, 6.34482408154679361777342971855, 7.05542574421894744726624605655, 7.727450857195703752638435295862, 9.06018636282300779272760015850, 9.666910294787784281909357509636, 10.61672518018151634040823308392, 11.38343464669336371911023455937, 11.86641154202505498907626100751, 12.80680321036606195089395484033, 14.29855435359930203094247494479, 15.04855066300233324363461432587, 16.02340946530874055343630309001, 16.60121884096936059207780958425, 17.67918948573505482384224884916, 17.76875849801852098392499666026, 18.659733555110052120267153913992, 19.7692417732397295157076347884, 20.54530868044394809686615413224, 21.22830121785935525485694310230, 22.09040949549441077188451386212

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