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Properties

Label 1-775-775.119-r1-0-0
Degree $1$
Conductor $775$
Sign $0.959 - 0.282i$
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 + (−0.978 + 0.207i)3-s + (0.309 − 0.951i)4-s + (−0.669 + 0.743i)6-s + (0.5 + 0.866i)7-s + (−0.309 − 0.951i)8-s + (0.913 − 0.406i)9-s + (0.104 + 0.994i)11-s + (−0.104 + 0.994i)12-s + (0.913 − 0.406i)13-s + (0.913 + 0.406i)14-s + (−0.809 − 0.587i)16-s + (0.669 − 0.743i)17-s + (0.5 − 0.866i)18-s + (0.669 − 0.743i)19-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (−0.978 + 0.207i)3-s + (0.309 − 0.951i)4-s + (−0.669 + 0.743i)6-s + (0.5 + 0.866i)7-s + (−0.309 − 0.951i)8-s + (0.913 − 0.406i)9-s + (0.104 + 0.994i)11-s + (−0.104 + 0.994i)12-s + (0.913 − 0.406i)13-s + (0.913 + 0.406i)14-s + (−0.809 − 0.587i)16-s + (0.669 − 0.743i)17-s + (0.5 − 0.866i)18-s + (0.669 − 0.743i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $0.959 - 0.282i$
Analytic conductor: \(83.2853\)
Root analytic conductor: \(83.2853\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 775,\ (1:\ ),\ 0.959 - 0.282i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.814240220 - 0.4064026965i\)
\(L(\frac12)\) \(\approx\) \(2.814240220 - 0.4064026965i\)
\(L(1)\) \(\approx\) \(1.423324202 - 0.3083953545i\)
\(L(1)\) \(\approx\) \(1.423324202 - 0.3083953545i\)

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 \( 1 + (-0.978 + 0.207i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.104 + 0.994i)T \)
13 \( 1 + (0.913 - 0.406i)T \)
17 \( 1 + (0.669 - 0.743i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (-0.809 + 0.587i)T \)
29 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (-0.104 + 0.994i)T \)
41 \( 1 + (0.913 - 0.406i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (0.669 + 0.743i)T \)
59 \( 1 + (-0.104 + 0.994i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (0.978 + 0.207i)T \)
71 \( 1 + (0.669 + 0.743i)T \)
73 \( 1 + (-0.104 - 0.994i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (-0.978 - 0.207i)T \)
89 \( 1 + (0.809 - 0.587i)T \)
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

−22.47790044957505546985510313231, −21.26618717420903936935083525773, −21.19350076273505785122554181836, −19.92445580803648319881858066439, −18.72980004876106697358570598247, −17.956646121812885224405322047233, −17.09464431657994069076788916377, −16.3824867264165401363914261286, −16.065379621559604781027877069213, −14.71214372275521719469093259442, −13.952228239340037646876112073885, −13.30644400169401075324424438094, −12.39425973071484958650983997701, −11.46921562041556729871324684987, −11.00711835728206248597152280367, −9.94515057294686685692745799579, −8.311336621550159654566451275995, −7.79755459144604923122099461526, −6.69502322797911154962885229119, −6.00275107945634045412110218003, −5.31660907362941825694781212832, −4.10633832726710319764148281364, −3.647426684809194640557797608691, −1.870499635937996558656924364, −0.70944881499905589560622765665, 0.91126016271530355298300066623, 1.83014665701115127151019890229, 3.08081778393779531115282562713, 4.17056715123277812476018447930, 5.15235537003055158105475525972, 5.5517443979258541600153283781, 6.585171311507951321453543127052, 7.55817610273852494288893434147, 9.12354170083358797030478356774, 9.86986626766097026329327443719, 10.7842182538399155975758396600, 11.59973901949499458379596000065, 12.08323823901026423305731135574, 12.85930593809273857461817642165, 13.84360778185224269425705535729, 14.853833502335330922439098969714, 15.61347665675482712385569418192, 16.0957432597054254990096298518, 17.4860231982847758075411343169, 18.21066474563891189915268420079, 18.69867677372107267217010642073, 20.04917870796252936559123708784, 20.688524819557905531340966158475, 21.442761276984012234515489801183, 22.21097617276072896807474302792

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