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Properties

Label 1-671-671.179-r0-0-0
Degree $1$
Conductor $671$
Sign $0.336 + 0.941i$
Analytic cond. $3.11611$
Root an. cond. $3.11611$
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  + (−0.5 + 0.866i)2-s + (−0.809 − 0.587i)3-s + (−0.5 − 0.866i)4-s + (0.913 − 0.406i)5-s + (0.913 − 0.406i)6-s + (−0.104 − 0.994i)7-s + 8-s + (0.309 + 0.951i)9-s + (−0.104 + 0.994i)10-s + (−0.104 + 0.994i)12-s + (0.669 + 0.743i)13-s + (0.913 + 0.406i)14-s + (−0.978 − 0.207i)15-s + (−0.5 + 0.866i)16-s + (−0.104 + 0.994i)17-s + (−0.978 − 0.207i)18-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.809 − 0.587i)3-s + (−0.5 − 0.866i)4-s + (0.913 − 0.406i)5-s + (0.913 − 0.406i)6-s + (−0.104 − 0.994i)7-s + 8-s + (0.309 + 0.951i)9-s + (−0.104 + 0.994i)10-s + (−0.104 + 0.994i)12-s + (0.669 + 0.743i)13-s + (0.913 + 0.406i)14-s + (−0.978 − 0.207i)15-s + (−0.5 + 0.866i)16-s + (−0.104 + 0.994i)17-s + (−0.978 − 0.207i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $0.336 + 0.941i$
Analytic conductor: \(3.11611\)
Root analytic conductor: \(3.11611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 671,\ (0:\ ),\ 0.336 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6548973107 + 0.4615335155i\)
\(L(\frac12)\) \(\approx\) \(0.6548973107 + 0.4615335155i\)
\(L(1)\) \(\approx\) \(0.6997796357 + 0.1626726674i\)
\(L(1)\) \(\approx\) \(0.6997796357 + 0.1626726674i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
61 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (0.913 - 0.406i)T \)
7 \( 1 + (-0.104 - 0.994i)T \)
13 \( 1 + (0.669 + 0.743i)T \)
17 \( 1 + (-0.104 + 0.994i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (-0.809 + 0.587i)T \)
29 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (-0.104 + 0.994i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (0.913 - 0.406i)T \)
47 \( 1 + (0.913 + 0.406i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
67 \( 1 + (-0.104 - 0.994i)T \)
71 \( 1 + (-0.978 + 0.207i)T \)
73 \( 1 + (0.669 - 0.743i)T \)
79 \( 1 + (-0.104 - 0.994i)T \)
83 \( 1 + (-0.104 - 0.994i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.104 + 0.994i)T \)
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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

−22.39059073239774675637254061010, −21.84811425252572391953963073911, −20.91079715305066965896439460155, −20.556059942776382669367025677481, −19.13289762500605154486315171786, −18.33830209163376021236849305526, −17.81530776922494026489217387838, −17.21419994973408755131635906343, −16.06946073428745399861352135225, −15.44221705934962022350479542049, −14.17532526893939780671932485577, −13.16874385798619063946767853510, −12.39774317896395056821062731017, −11.43918902796945852692228705821, −10.87290097478469450246299974816, −9.92240342039872202881943609114, −9.36537780867633194468407836352, −8.528065770418825478565053769228, −7.093856234158739862713300329405, −5.97180599630570098286291099367, −5.27652507235825140606893654969, −4.1079272604910691776304823226, −2.90618506425681742978776864767, −2.138067937558637308764562119178, −0.57513025266128719050632906156, 1.26316296515254889840998993971, 1.683402683018721494229492217018, 3.97544458010300521852798398149, 4.96844954643190943739162051678, 5.99032253823346045636793681658, 6.42725484062622582009401360832, 7.39270918051717306644542112988, 8.29681565031510933898399468533, 9.299262265820192037339055857430, 10.42054695046997171185489890386, 10.70301088687578422408673789979, 12.19124270683774010799836794523, 13.16446718465072678046820351202, 13.81114698754703422156063121025, 14.45003093255104900197277764446, 16.03102136988197189427409999893, 16.45889115861616013375305427427, 17.25464517717625381766964926154, 17.71313073408919648467979899016, 18.581440577553629108036371596324, 19.38051907838703338707986855113, 20.3356792873848306175244160116, 21.47199272562006184552639044284, 22.329204365606399819714035875901, 23.29670134615996739259911614648

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