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Properties

Label 1-3311-3311.1006-r1-0-0
Degree $1$
Conductor $3311$
Sign $0.194 - 0.980i$
Analytic cond. $355.816$
Root an. cond. $355.816$
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  + (−0.999 + 0.0299i)2-s + (0.925 − 0.379i)3-s + (0.998 − 0.0598i)4-s + (−0.753 + 0.657i)5-s + (−0.913 + 0.406i)6-s + (−0.995 + 0.0896i)8-s + (0.712 − 0.701i)9-s + (0.733 − 0.680i)10-s + (0.900 − 0.433i)12-s + (0.599 − 0.800i)13-s + (−0.447 + 0.894i)15-s + (0.992 − 0.119i)16-s + (0.393 − 0.919i)17-s + (−0.691 + 0.722i)18-s + (−0.936 − 0.351i)19-s + (−0.712 + 0.701i)20-s + ⋯
L(s)  = 1  + (−0.999 + 0.0299i)2-s + (0.925 − 0.379i)3-s + (0.998 − 0.0598i)4-s + (−0.753 + 0.657i)5-s + (−0.913 + 0.406i)6-s + (−0.995 + 0.0896i)8-s + (0.712 − 0.701i)9-s + (0.733 − 0.680i)10-s + (0.900 − 0.433i)12-s + (0.599 − 0.800i)13-s + (−0.447 + 0.894i)15-s + (0.992 − 0.119i)16-s + (0.393 − 0.919i)17-s + (−0.691 + 0.722i)18-s + (−0.936 − 0.351i)19-s + (−0.712 + 0.701i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $0.194 - 0.980i$
Analytic conductor: \(355.816\)
Root analytic conductor: \(355.816\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (1006, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3311,\ (1:\ ),\ 0.194 - 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.528282960 - 1.255544567i\)
\(L(\frac12)\) \(\approx\) \(1.528282960 - 1.255544567i\)
\(L(1)\) \(\approx\) \(0.9368804119 - 0.1811965720i\)
\(L(1)\) \(\approx\) \(0.9368804119 - 0.1811965720i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.999 + 0.0299i)T \)
3 \( 1 + (0.925 - 0.379i)T \)
5 \( 1 + (-0.753 + 0.657i)T \)
13 \( 1 + (0.599 - 0.800i)T \)
17 \( 1 + (0.393 - 0.919i)T \)
19 \( 1 + (-0.936 - 0.351i)T \)
23 \( 1 + (0.623 + 0.781i)T \)
29 \( 1 + (0.791 - 0.611i)T \)
31 \( 1 + (0.999 - 0.0299i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (0.691 + 0.722i)T \)
47 \( 1 + (0.163 + 0.986i)T \)
53 \( 1 + (0.193 - 0.981i)T \)
59 \( 1 + (-0.575 + 0.817i)T \)
61 \( 1 + (0.999 + 0.0299i)T \)
67 \( 1 + (-0.988 - 0.149i)T \)
71 \( 1 + (0.712 + 0.701i)T \)
73 \( 1 + (-0.858 - 0.512i)T \)
79 \( 1 + (-0.104 - 0.994i)T \)
83 \( 1 + (0.999 + 0.0299i)T \)
89 \( 1 + (0.900 + 0.433i)T \)
97 \( 1 + (0.963 + 0.266i)T \)
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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

−19.00766947907879374235509363469, −18.47769905680384532981586882907, −17.254966228703216464040136417345, −16.72507210549279417602145227765, −16.14447913739238768917975316571, −15.4817498121916671836116247780, −14.92505186837766434304055858926, −14.19855514822512028761639850534, −13.15473695473566333862996246928, −12.46268427022072175121701629233, −11.79789962776422841702449107561, −10.80098856229216234535940188878, −10.38911720127244385901212922631, −9.462683019848661750017908840771, −8.613436662261474312221751328589, −8.53399017630287533451453122340, −7.76642016642799439171307247958, −6.895036114695328186529148724723, −6.1540055607798654511548313263, −4.878480406924782885267995452801, −4.100860268163535393568079633, −3.42644451363125705971870515451, −2.48912072862281283708942182640, −1.58867005501544570561472472136, −0.85989542200456405634259211171, 0.48672730811766777509352591427, 1.10019743066303406397647195947, 2.35103618216083070700595400138, 2.88579938241003473461067045176, 3.52973257499319699230477385513, 4.55176609772374607221953268271, 5.95904087378526918384662159403, 6.622876144348697405129526512106, 7.40349539372500598163011971435, 7.8531408080951875390225923705, 8.48937562684734271309753791369, 9.21175768826561109819303771090, 9.99236581767479179584607699092, 10.68671968433829364058624816957, 11.424549965987018941229292124703, 12.09443007186184558167319100863, 12.9249284981124992274048272966, 13.737352624316551030680654715413, 14.67046676544617732474048439074, 15.10658517426214251430150364205, 15.80100604717487900422668305963, 16.249000912367641788227254481838, 17.61990035281827754030090188655, 17.81619481944255821172579112999, 18.72070869600615955400615839445

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