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Properties

Label 1-671-671.101-r0-0-0
Degree $1$
Conductor $671$
Sign $-0.699 + 0.714i$
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  + (−0.994 − 0.104i)2-s + (−0.309 + 0.951i)3-s + (0.978 + 0.207i)4-s + (−0.913 − 0.406i)5-s + (0.406 − 0.913i)6-s + (0.743 + 0.669i)7-s + (−0.951 − 0.309i)8-s + (−0.809 − 0.587i)9-s + (0.866 + 0.5i)10-s + (−0.5 + 0.866i)12-s + (0.104 − 0.994i)13-s + (−0.669 − 0.743i)14-s + (0.669 − 0.743i)15-s + (0.913 + 0.406i)16-s + (−0.994 + 0.104i)17-s + (0.743 + 0.669i)18-s + ⋯
L(s)  = 1  + (−0.994 − 0.104i)2-s + (−0.309 + 0.951i)3-s + (0.978 + 0.207i)4-s + (−0.913 − 0.406i)5-s + (0.406 − 0.913i)6-s + (0.743 + 0.669i)7-s + (−0.951 − 0.309i)8-s + (−0.809 − 0.587i)9-s + (0.866 + 0.5i)10-s + (−0.5 + 0.866i)12-s + (0.104 − 0.994i)13-s + (−0.669 − 0.743i)14-s + (0.669 − 0.743i)15-s + (0.913 + 0.406i)16-s + (−0.994 + 0.104i)17-s + (0.743 + 0.669i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $-0.699 + 0.714i$
Analytic conductor: \(3.11611\)
Root analytic conductor: \(3.11611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 671,\ (0:\ ),\ -0.699 + 0.714i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1700183114 + 0.4046512996i\)
\(L(\frac12)\) \(\approx\) \(0.1700183114 + 0.4046512996i\)
\(L(1)\) \(\approx\) \(0.4903040830 + 0.1772732263i\)
\(L(1)\) \(\approx\) \(0.4903040830 + 0.1772732263i\)

Euler product

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

−22.74279552577054034030128026415, −21.567124095412475480689885562364, −20.37364546352355741867022554668, −19.7997582102107170666828604678, −19.07529961032156907532275545254, −18.398748733468717821040912975514, −17.58135489448514660961482709742, −17.02371583814430817543950884979, −16.003783446081629014061144023077, −15.207984329077705357701933152191, −14.16401326173167372048647564771, −13.35447212604856726052734844445, −11.93286472067069910698113759012, −11.36556606642186154777508751220, −11.05112131679697670830448691864, −9.71063535124498989592081117461, −8.533610907083874815988486322667, −7.873499610230256039981956151347, −7.04912389392467587223938653047, −6.65410041889947110921861080538, −5.242487394617054917140572704607, −3.93545437544424895999700655657, −2.52916720288641990666400830011, −1.567754602466811576239697995648, −0.354182256118631771092591723932, 1.16429825846683046261784478619, 2.7013136389312640365170229056, 3.66763717123363523464995848078, 4.84700265872589156754314544545, 5.67551302372866549762474194998, 6.92629948200225666890975173354, 8.19664768440261749765407586880, 8.537114300057042125297563241503, 9.42911684780105895733332936470, 10.63321790750472077002246122709, 10.999591152722854715834291790725, 12.0441502432616242428773114143, 12.48212231310297494201651571153, 14.36712795630693507984536167495, 15.28097992339565625993384117092, 15.70520617994103975681605900853, 16.43423200557653421704941835879, 17.37672317041214036154651211313, 18.044333600658412223184819141963, 18.9102199981283669204325979135, 20.1150351037548392737177141433, 20.37081428034816536006620174577, 21.20750354517446634977159592622, 22.15761763740117912866294877105, 22.95608055971393898656773818602

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