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Properties

Label 1-77-77.52-r0-0-0
Degree $1$
Conductor $77$
Sign $0.394 - 0.918i$
Analytic cond. $0.357586$
Root an. cond. $0.357586$
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  + (−0.669 − 0.743i)2-s + (−0.913 − 0.406i)3-s + (−0.104 + 0.994i)4-s + (0.978 + 0.207i)5-s + (0.309 + 0.951i)6-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)12-s + (0.309 − 0.951i)13-s + (−0.809 − 0.587i)15-s + (−0.978 − 0.207i)16-s + (0.669 − 0.743i)17-s + (0.104 − 0.994i)18-s + (−0.104 − 0.994i)19-s + (−0.309 + 0.951i)20-s + ⋯
L(s)  = 1  + (−0.669 − 0.743i)2-s + (−0.913 − 0.406i)3-s + (−0.104 + 0.994i)4-s + (0.978 + 0.207i)5-s + (0.309 + 0.951i)6-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)12-s + (0.309 − 0.951i)13-s + (−0.809 − 0.587i)15-s + (−0.978 − 0.207i)16-s + (0.669 − 0.743i)17-s + (0.104 − 0.994i)18-s + (−0.104 − 0.994i)19-s + (−0.309 + 0.951i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.394 - 0.918i$
Analytic conductor: \(0.357586\)
Root analytic conductor: \(0.357586\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 77,\ (0:\ ),\ 0.394 - 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5200810637 - 0.3426292388i\)
\(L(\frac12)\) \(\approx\) \(0.5200810637 - 0.3426292388i\)
\(L(1)\) \(\approx\) \(0.6371066084 - 0.2775059365i\)
\(L(1)\) \(\approx\) \(0.6371066084 - 0.2775059365i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.669 - 0.743i)T \)
3 \( 1 + (-0.913 - 0.406i)T \)
5 \( 1 + (0.978 + 0.207i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (0.669 - 0.743i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (0.913 - 0.406i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (0.104 + 0.994i)T \)
53 \( 1 + (-0.978 + 0.207i)T \)
59 \( 1 + (0.104 - 0.994i)T \)
61 \( 1 + (-0.978 - 0.207i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.104 + 0.994i)T \)
79 \( 1 + (-0.669 - 0.743i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
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

−32.04056911209904780308309482140, −30.03734401126547555169325158227, −28.74772858639219123192296626907, −28.43993232086043156581091025467, −27.1290157595549344852854038296, −26.16633492207086621930197288048, −25.07784814234474729086077663690, −23.96685289453082124617080289671, −23.02024732772653358958388383828, −21.72902667951501779024423385275, −20.69938118548549724952638779903, −18.903618503259431910732999483360, −17.98205143848762077994705843621, −16.89408028162634199386321784866, −16.387733146259587934262925862416, −14.93386218336342855567923544551, −13.734288209342602138631068822928, −12.057865750006975452235328536953, −10.46535514734965077179147337172, −9.80036821169651843471528557128, −8.44014927760579056604488894799, −6.56903064878054751189914479161, −5.86117417954088765378325226470, −4.52168944180130289707554079488, −1.51748156736491366199864976149, 1.226667632795239744919217138014, 2.83286506441352246292378987142, 5.03118080908245419444905886997, 6.49835278750329168216903492468, 7.86236294329531123574903993983, 9.56501116828588345954546096830, 10.51046905943477602619180911902, 11.57565514870907457338891952663, 12.81408543844951607558137790048, 13.71473037378921118561994183769, 15.89800302967016592659856927675, 17.17215165978999539131729656980, 17.857865621843145734112327877957, 18.66269285454894993960267675987, 20.04520818862382029292271070211, 21.37260739847188662083132098915, 22.1307024262426401400249186566, 23.23751094698265507427040242029, 24.8903542396174203469670754660, 25.67413928138608022825007220467, 27.10213467459811594631450931317, 28.10536031708068914022735378826, 28.923459617218046623799495153816, 29.921207369312005637179499946690, 30.34035709540946143939848768724

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