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Properties

Label 1-77-77.2-r1-0-0
Degree $1$
Conductor $77$
Sign $0.999 + 0.0413i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.999 + 0.0413i$
Analytic conductor: \(8.27479\)
Root analytic conductor: \(8.27479\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 77,\ (1:\ ),\ 0.999 + 0.0413i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.251266370 + 0.04659604814i\)
\(L(\frac12)\) \(\approx\) \(2.251266370 + 0.04659604814i\)
\(L(1)\) \(\approx\) \(1.498402727 - 0.1413716346i\)
\(L(1)\) \(\approx\) \(1.498402727 - 0.1413716346i\)

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

−31.207210327987978488668842690853, −30.110345402649798439892923014649, −28.97345397555375924623370938685, −27.56475618503838174118037578745, −26.22963162036450171505915222196, −25.37349466850366118527242014407, −24.767504529970548512347284194666, −23.75338362835300443949938053628, −22.572637763272856117082712151629, −21.17498266785572947498203039862, −20.05443061112682363341189051477, −18.35108684442796904129676650686, −17.9208637954264177159000449005, −16.53631980727868584508690602585, −15.28771555214577840372637713511, −13.86033624399808846697747541059, −13.50841151388209643882894299324, −12.19879006266049391013034606139, −9.80558248802568372072154476534, −8.81058179120644084335464802873, −7.67443176994234681418999322979, −6.39459928550965748784628097927, −5.24001688395738564355787655872, −3.251295101468902437200171426505, −1.16516552451578813067856332630, 1.81470676096514541535317368083, 3.108943844379974919248144134763, 4.41551453043681888151884653747, 5.95897081106166522923643912631, 8.29329981995462965908975166992, 9.48104522007468944025756757631, 10.283909304573664965284519269452, 11.40028719708086034170650677385, 13.19398400678703895436873270603, 13.979494611147309883417695654083, 14.98785685841571180234051172474, 16.66342617994356019511955865763, 18.05742444410418972854212299957, 19.090509776141324821779712668635, 20.29831562889206396352988518972, 21.213844427744360310328603944934, 21.889451028496555182096312143017, 22.95635564991275309795003744775, 24.6559135843507593997065260164, 26.16878695402520888675146099834, 26.51354596499764812775969804940, 28.08581015808625623097830085642, 28.70998989422990517666257497504, 30.21496415072832001247827802635, 30.75037878403591246700101997783

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