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Properties

Label 1-3724-3724.139-r0-0-0
Degree $1$
Conductor $3724$
Sign $-0.740 - 0.671i$
Analytic cond. $17.2941$
Root an. cond. $17.2941$
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  + (0.980 − 0.198i)3-s + (0.318 − 0.947i)5-s + (0.921 − 0.388i)9-s + (0.733 + 0.680i)11-s + (−0.456 − 0.889i)13-s + (0.124 − 0.992i)15-s + (−0.270 − 0.962i)17-s + (−0.698 − 0.715i)23-s + (−0.797 − 0.603i)25-s + (0.826 − 0.563i)27-s + (−0.583 − 0.811i)29-s + (−0.5 − 0.866i)31-s + (0.853 + 0.521i)33-s + (−0.900 + 0.433i)37-s + (−0.623 − 0.781i)39-s + ⋯
L(s)  = 1  + (0.980 − 0.198i)3-s + (0.318 − 0.947i)5-s + (0.921 − 0.388i)9-s + (0.733 + 0.680i)11-s + (−0.456 − 0.889i)13-s + (0.124 − 0.992i)15-s + (−0.270 − 0.962i)17-s + (−0.698 − 0.715i)23-s + (−0.797 − 0.603i)25-s + (0.826 − 0.563i)27-s + (−0.583 − 0.811i)29-s + (−0.5 − 0.866i)31-s + (0.853 + 0.521i)33-s + (−0.900 + 0.433i)37-s + (−0.623 − 0.781i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $-0.740 - 0.671i$
Analytic conductor: \(17.2941\)
Root analytic conductor: \(17.2941\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (0:\ ),\ -0.740 - 0.671i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7685096227 - 1.992000187i\)
\(L(\frac12)\) \(\approx\) \(0.7685096227 - 1.992000187i\)
\(L(1)\) \(\approx\) \(1.320103553 - 0.5996285791i\)
\(L(1)\) \(\approx\) \(1.320103553 - 0.5996285791i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.980 - 0.198i)T \)
5 \( 1 + (0.318 - 0.947i)T \)
11 \( 1 + (0.733 + 0.680i)T \)
13 \( 1 + (-0.456 - 0.889i)T \)
17 \( 1 + (-0.270 - 0.962i)T \)
23 \( 1 + (-0.698 - 0.715i)T \)
29 \( 1 + (-0.583 - 0.811i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.900 + 0.433i)T \)
41 \( 1 + (-0.980 + 0.198i)T \)
43 \( 1 + (0.853 + 0.521i)T \)
47 \( 1 + (-0.998 - 0.0498i)T \)
53 \( 1 + (0.698 + 0.715i)T \)
59 \( 1 + (0.878 - 0.478i)T \)
61 \( 1 + (-0.995 + 0.0995i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (-0.995 - 0.0995i)T \)
73 \( 1 + (-0.456 + 0.889i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.733 + 0.680i)T \)
89 \( 1 + (0.797 + 0.603i)T \)
97 \( 1 + (-0.173 + 0.984i)T \)
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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

−19.19016964146183538605647488568, −18.30290683456719377802694614332, −17.62096350457063886256126918706, −16.77755850063414577534287839988, −16.07475717173546225290702626319, −15.30086298230869087871957484380, −14.54852085952459474237210569617, −14.2682942164757552106691089559, −13.62139214306794929516400006508, −12.872438565627841489230837177844, −11.90422415861035888426544271751, −11.19516445566940651296418213944, −10.37255251648095767655296210348, −9.89803076427137605746195386412, −8.8839862399978329125820951909, −8.69581837236694039270613172648, −7.45285924428601069244812040562, −7.04625426124236511769866321466, −6.24068978950589973080391558410, −5.393630603930667228204464027357, −4.24343282113755480868661410201, −3.6078085152596469360550686346, −3.0804015223040087502579653508, −1.902792845046863537743362192308, −1.667426265898920024727764847220, 0.46230554268624324643083326995, 1.53397321612617022931938212749, 2.18664860007804426350518366413, 3.00152948353126889883693677760, 4.05085662365316161577397110121, 4.5701979353631980537984580402, 5.43811057118303680620310471878, 6.3742975213899993441001006190, 7.22110545536958276297609529071, 7.8987337052306697620224963167, 8.53734703346301558536902580262, 9.370044633180268149787020164309, 9.6818462372491081789205452870, 10.45776397998893345157856026853, 11.77116105706748863662138706011, 12.20870481890976978737559464313, 13.016771655083142115340252832929, 13.44522602203605473817393119990, 14.244235371740480102532030579791, 14.89169923382720578256701548285, 15.533027540725238308984871302030, 16.2700668781419240125798420242, 17.05516565522501725674823458608, 17.70250975248254533404301100788, 18.34670836671193441955247759987

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