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Properties

Label 1-77-77.27-r1-0-0
Degree $1$
Conductor $77$
Sign $0.935 - 0.352i$
Analytic cond. $8.27479$
Root an. cond. $8.27479$
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 + (−0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (0.809 + 0.587i)6-s + (0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s − 10-s − 12-s + (0.809 − 0.587i)13-s + (0.309 − 0.951i)15-s + (−0.809 − 0.587i)16-s + (0.809 + 0.587i)17-s + (0.309 − 0.951i)18-s + (−0.309 − 0.951i)19-s + (0.809 − 0.587i)20-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (0.809 + 0.587i)6-s + (0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s − 10-s − 12-s + (0.809 − 0.587i)13-s + (0.309 − 0.951i)15-s + (−0.809 − 0.587i)16-s + (0.809 + 0.587i)17-s + (0.309 − 0.951i)18-s + (−0.309 − 0.951i)19-s + (0.809 − 0.587i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.935 - 0.352i$
Analytic conductor: \(8.27479\)
Root analytic conductor: \(8.27479\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 77,\ (1:\ ),\ 0.935 - 0.352i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.143628323 - 0.2080575902i\)
\(L(\frac12)\) \(\approx\) \(1.143628323 - 0.2080575902i\)
\(L(1)\) \(\approx\) \(0.8408689849 - 0.03850472566i\)
\(L(1)\) \(\approx\) \(0.8408689849 - 0.03850472566i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 + T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 - T \)
97 \( 1 + (0.809 - 0.587i)T \)
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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

−31.0220350705373066737310801196, −29.48067417311086922517288406028, −28.82303449247007220836582865578, −27.88174661697620142219442362486, −27.04661599863471794595195682540, −25.831710269202203228206864145134, −25.09799166618416766819208327394, −23.263070558992003163293086773461, −21.90421611804432634331539913870, −20.992412110723878767450417264936, −20.515395486565823834446005700496, −18.90054064037993331690103618599, −17.6793315724272640203010818700, −16.700885711614448760375008973593, −16.014550602416313814098008659110, −14.205410670274665819532065676107, −12.68221218349272029227328242458, −11.44798993251795443306792662152, −10.2713417323658349602893617409, −9.389573118774003644612996694402, −8.401848633306559213227354425640, −6.341183253011033851140828684001, −4.75190656237086900072548356975, −3.17819665792546424090336319627, −1.23182753820516844374618733760, 0.96465532198887001435257401458, 2.48212295941046432136210154003, 5.54590395183133903701703137792, 6.3975435696178778641933436552, 7.52107039970816027673152663554, 8.79579719187064015022848482775, 10.3121140632686224470846463879, 11.285532789887579610731650655534, 13.08172386382919047117883274058, 14.127959091358391932630238962747, 15.37013233647905810539566890306, 16.99195253874380967408532151966, 17.64346567685946162044269023725, 18.62654594466980353362507434157, 19.42426177922738941617704658238, 20.97719044652674181344707567032, 22.65770363604683125422513179021, 23.52360036155062422879406979006, 24.75717159691271969661857980112, 25.49639112568298736627136228352, 26.338841269895710999083861130384, 27.88679535796053365562136257351, 28.709645766484904512992613084437, 29.776205668248861807611844847938, 30.52162968969400477537506005998

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