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Properties

Label 1-77-77.38-r1-0-0
Degree $1$
Conductor $77$
Sign $0.286 - 0.958i$
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  + (−0.104 − 0.994i)2-s + (−0.669 + 0.743i)3-s + (−0.978 + 0.207i)4-s + (−0.913 + 0.406i)5-s + (0.809 + 0.587i)6-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + (0.5 + 0.866i)10-s + (0.5 − 0.866i)12-s + (0.809 − 0.587i)13-s + (0.309 − 0.951i)15-s + (0.913 − 0.406i)16-s + (0.104 − 0.994i)17-s + (−0.978 + 0.207i)18-s + (0.978 + 0.207i)19-s + (0.809 − 0.587i)20-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (−0.669 + 0.743i)3-s + (−0.978 + 0.207i)4-s + (−0.913 + 0.406i)5-s + (0.809 + 0.587i)6-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + (0.5 + 0.866i)10-s + (0.5 − 0.866i)12-s + (0.809 − 0.587i)13-s + (0.309 − 0.951i)15-s + (0.913 − 0.406i)16-s + (0.104 − 0.994i)17-s + (−0.978 + 0.207i)18-s + (0.978 + 0.207i)19-s + (0.809 − 0.587i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.286 - 0.958i$
Analytic conductor: \(8.27479\)
Root analytic conductor: \(8.27479\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 77,\ (1:\ ),\ 0.286 - 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6465580891 - 0.4815747843i\)
\(L(\frac12)\) \(\approx\) \(0.6465580891 - 0.4815747843i\)
\(L(1)\) \(\approx\) \(0.6494262755 - 0.2059892683i\)
\(L(1)\) \(\approx\) \(0.6494262755 - 0.2059892683i\)

Euler product

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

−31.12219732846546566053551285827, −30.52952276104799010226678647496, −28.67154024477893313672676779045, −28.05981560704765574820606829851, −26.90579713061971986176914689065, −25.691784502463078973354530587260, −24.4447042283942796995274217839, −23.77623565608549273811034352175, −23.03560977470562140929281174966, −21.86623299529852623716230427455, −19.902229464621936743497918082620, −18.79057735252568087515525047938, −17.921631170641704452853844106418, −16.57416404010097044449249918152, −16.01973811843034091253404690185, −14.49931171996894883273959485724, −13.18206637392538241278037478453, −12.16694484295162503783678183694, −10.774841718156788095060721143736, −8.833968814775789819149466864988, −7.7975821652373573521395426415, −6.70851619070142582558867816695, −5.44926412560395810572482387572, −4.06167025399182612765146742425, −1.03682618046894381457000602721, 0.613133482446388383627320367945, 3.157583836049092538427981563157, 4.168766207322261870836497359760, 5.60580079608861178945707152066, 7.67760779658338507885083394054, 9.22883774373711414298307974071, 10.417774697561183903051901560256, 11.40259499945233701654919457006, 12.1113495109055197732422058715, 13.764508493478388604976105964353, 15.263072126424306144540528041900, 16.30847916222997920212914226639, 17.78804805940126100627384542274, 18.63820286456144659009864463463, 20.06045152748018098023175953806, 20.83682552733400365745480933416, 22.20279027244962585141833903579, 22.805660375060516723436701859056, 23.75051333714338530104236159540, 25.87344863273161382417823767137, 27.062982310126908112955633052412, 27.52044923379005514212980500414, 28.563308755965212589901734734414, 29.60441892213044176882935851214, 30.7152792912074546196196392071

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