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Properties

Label 1-77-77.17-r0-0-0
Degree $1$
Conductor $77$
Sign $-0.782 - 0.622i$
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  + (0.104 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $-0.782 - 0.622i$
Analytic conductor: \(0.357586\)
Root analytic conductor: \(0.357586\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 77,\ (0:\ ),\ -0.782 - 0.622i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1226950679 + 0.3511286066i\)
\(L(\frac12)\) \(\approx\) \(-0.1226950679 + 0.3511286066i\)
\(L(1)\) \(\approx\) \(0.3272553678 + 0.4587057206i\)
\(L(1)\) \(\approx\) \(0.3272553678 + 0.4587057206i\)

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

−30.41744985862039923895407922863, −29.59913052311906155662101743854, −28.56919227828043923484176028198, −27.70135740049758784147028068908, −26.84853658552083510482659842172, −24.90544663947560325893002085091, −23.81539020702033754431881718711, −22.94970454446800460615169487392, −22.10155790889310498336140989440, −20.54924402116639417415433718517, −19.59157998960460313997574232703, −18.71578287011481195209877691419, −17.59594938252360160092776104101, −16.40517879352628457890510350491, −14.6960027932465767293734673534, −13.17625675491781081396558123225, −12.32240055879339962641219867029, −11.51262593668324876113620344397, −10.33626417667269482302335223532, −8.61695049597304819532942938006, −7.37877744830931553683305296653, −5.470744227307850620940774120219, −4.24701597376632252383815978859, −2.36816056014424033241386735469, −0.43848158120631132866764026584, 3.70030512299410343531230709086, 4.676868133353352725910319757672, 6.14170461017547026166533397239, 7.31558767720746658554243146486, 8.75318597444822468941823126724, 10.12720929337523328390176863726, 11.49487317795740836713636160868, 12.73944595128524385175767609207, 14.624661599788439498836812524922, 15.20859186420637194141544998292, 16.35774171131141013908194527800, 17.18138161006737324924908036010, 18.44819395441430824906792818551, 19.735192270341948431214124200327, 21.62293101319792692831996549831, 22.21594039936161846536311365967, 23.55973495870992065529327344383, 23.8690807796881950269936524642, 25.7052950695354949286363112538, 26.602057473281608858841176336555, 27.42433912499378830723868560875, 28.25331312530179917013721858747, 29.86967130395994092091347907166, 31.29278252352979224035553322082, 32.01767021116261060295324537358

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