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Properties

Label 1-77-77.59-r1-0-0
Degree $1$
Conductor $77$
Sign $-0.624 + 0.781i$
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  + (−0.978 − 0.207i)2-s + (0.104 − 0.994i)3-s + (0.913 + 0.406i)4-s + (−0.669 − 0.743i)5-s + (−0.309 + 0.951i)6-s + (−0.809 − 0.587i)8-s + (−0.978 − 0.207i)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 + (0.669 + 0.743i)16-s + (0.978 − 0.207i)17-s + (0.913 + 0.406i)18-s + (−0.913 + 0.406i)19-s + (−0.309 − 0.951i)20-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (0.104 − 0.994i)3-s + (0.913 + 0.406i)4-s + (−0.669 − 0.743i)5-s + (−0.309 + 0.951i)6-s + (−0.809 − 0.587i)8-s + (−0.978 − 0.207i)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 + (0.669 + 0.743i)16-s + (0.978 − 0.207i)17-s + (0.913 + 0.406i)18-s + (−0.913 + 0.406i)19-s + (−0.309 − 0.951i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $-0.624 + 0.781i$
Analytic conductor: \(8.27479\)
Root analytic conductor: \(8.27479\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 77,\ (1:\ ),\ -0.624 + 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1151294862 - 0.2393518603i\)
\(L(\frac12)\) \(\approx\) \(-0.1151294862 - 0.2393518603i\)
\(L(1)\) \(\approx\) \(0.4163337123 - 0.2930042850i\)
\(L(1)\) \(\approx\) \(0.4163337123 - 0.2930042850i\)

Euler product

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

−31.83755547201695287710628229696, −30.56001036603616863009962130670, −29.31545233808577118241199399070, −27.98482079520296381984303566631, −27.44974489474998936928093201073, −26.18570263733253070329129896992, −25.97562911304579747886301244422, −24.23703166289592352321642089298, −23.05637418238028760706132997749, −21.74170608501481308397777562504, −20.62217673033795086002125674035, −19.45120460443461052727469973693, −18.64949369748002627808749641023, −17.091857387319626984340266293809, −16.244987272577396810237675201176, −15.102285681231770018981871289330, −14.41497923742296551430684475224, −11.8833287636447174090420440532, −10.9010291535842373775791005266, −9.93776023597150132771095085249, −8.716998706615419695995020535211, −7.48370415622714958129309334044, −6.05018598536805890984049220299, −4.13023738987600501882676080644, −2.54180706652128659531365323953, 0.1723106175390166654072120125, 1.57656629180911972297913377485, 3.32781154291160252868423602809, 5.72749105879624206628640543294, 7.42081899179604931759192157120, 8.07367609385282539720023899955, 9.281947150487250093204122569404, 10.94219587603111149695659479081, 12.2067221991621554213436275569, 12.82508133774253752710043451296, 14.73737554947154126119879663085, 16.15969993241783000134310447664, 17.19745133609336529148914605076, 18.230509718397648101674782944648, 19.378848354933436454134534477394, 19.99098481277878155356646500976, 21.11807003492511941507130367933, 23.02483407273507932829057173517, 24.1028760660918591707044247605, 25.02025033940084872183394979925, 25.87257146249783772340919687025, 27.422379125901819680294003206916, 28.009847560651749131602559006685, 29.32500934469724953555833412130, 29.98477089830405885374150233153

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