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Properties

Label 1-3381-3381.617-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.502 - 0.864i$
Analytic cond. $15.7012$
Root an. cond. $15.7012$
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 + 0.0407i)2-s + (0.996 + 0.0815i)4-s + (−0.768 − 0.639i)5-s + (0.992 + 0.122i)8-s + (−0.742 − 0.670i)10-s + (0.182 − 0.983i)11-s + (−0.377 − 0.925i)13-s + (0.986 + 0.162i)16-s + (0.996 − 0.0815i)17-s + (0.142 + 0.989i)19-s + (−0.714 − 0.699i)20-s + (0.222 − 0.974i)22-s + (0.182 + 0.983i)25-s + (−0.339 − 0.940i)26-s + (−0.996 + 0.0815i)29-s + ⋯
L(s)  = 1  + (0.999 + 0.0407i)2-s + (0.996 + 0.0815i)4-s + (−0.768 − 0.639i)5-s + (0.992 + 0.122i)8-s + (−0.742 − 0.670i)10-s + (0.182 − 0.983i)11-s + (−0.377 − 0.925i)13-s + (0.986 + 0.162i)16-s + (0.996 − 0.0815i)17-s + (0.142 + 0.989i)19-s + (−0.714 − 0.699i)20-s + (0.222 − 0.974i)22-s + (0.182 + 0.983i)25-s + (−0.339 − 0.940i)26-s + (−0.996 + 0.0815i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.502 - 0.864i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (617, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.502 - 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.753165319 - 1.584777196i\)
\(L(\frac12)\) \(\approx\) \(2.753165319 - 1.584777196i\)
\(L(1)\) \(\approx\) \(1.827963437 - 0.3812266687i\)
\(L(1)\) \(\approx\) \(1.827963437 - 0.3812266687i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.999 + 0.0407i)T \)
5 \( 1 + (-0.768 - 0.639i)T \)
11 \( 1 + (0.182 - 0.983i)T \)
13 \( 1 + (-0.377 - 0.925i)T \)
17 \( 1 + (0.996 - 0.0815i)T \)
19 \( 1 + (0.142 + 0.989i)T \)
29 \( 1 + (-0.996 + 0.0815i)T \)
31 \( 1 + (0.841 + 0.540i)T \)
37 \( 1 + (0.979 + 0.202i)T \)
41 \( 1 + (0.768 + 0.639i)T \)
43 \( 1 + (0.992 - 0.122i)T \)
47 \( 1 + (-0.623 - 0.781i)T \)
53 \( 1 + (-0.0611 - 0.998i)T \)
59 \( 1 + (-0.742 - 0.670i)T \)
61 \( 1 + (-0.339 + 0.940i)T \)
67 \( 1 + (0.654 - 0.755i)T \)
71 \( 1 + (0.591 + 0.806i)T \)
73 \( 1 + (-0.862 - 0.505i)T \)
79 \( 1 + (0.959 + 0.281i)T \)
83 \( 1 + (-0.452 - 0.891i)T \)
89 \( 1 + (0.101 - 0.994i)T \)
97 \( 1 + (-0.415 - 0.909i)T \)
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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

−19.18859995523133482398038100574, −18.42022301260137518556453087278, −17.40058632570730848879678442580, −16.72552057001715168424102790662, −15.92819692621634439875933368107, −15.33244424726518634075703099372, −14.69492891364829009013670415451, −14.26048367029494761005172975327, −13.44170068831393644805982246641, −12.48084096755854186622899305538, −12.115682197859745818473486447550, −11.318768089688798066486729793212, −10.86310012162230895218999730137, −9.862004025337504749693310047136, −9.23518518335276776171537773242, −7.75951080934058264767192736115, −7.532499066534828688914596385690, −6.72378763406783294288390273636, −6.07846289966965794408124613002, −5.032356751918293928787178531836, −4.31171406953930400895761921832, −3.83853585812750237118456311925, −2.77823867206112541156166969160, −2.27286387283420430318515519931, −1.10226913018628462941270960177, 0.72607391304862903581688240970, 1.57427294115520513547103349199, 2.89105938195278725776010548160, 3.415747870073564875130317292113, 4.12020555933625065727561749796, 5.017260618494142945968450246485, 5.628422970716107441642556075092, 6.25093086752923836162143753984, 7.43557966075596013152468281087, 7.87516522969592747142214098512, 8.51003417789782740646130772719, 9.66776022080018567655325802266, 10.442470576029501717810004369049, 11.32772872861239416865704417506, 11.771134213136877908437099220618, 12.640835310277328259995434009518, 12.90293607762928906828521779157, 13.92642558309973118672717489088, 14.504705111840769123179595527753, 15.210174416983627841568468011179, 15.873619885198526029228863388095, 16.67887766586845116777748181910, 16.76465760758045272902515220839, 18.02794555357099913904631006956, 19.03116064913916379206657158108

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