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Properties

Label 1-3724-3724.1399-r1-0-0
Degree $1$
Conductor $3724$
Sign $-0.999 + 0.0124i$
Analytic cond. $400.199$
Root an. cond. $400.199$
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  + (−0.826 + 0.563i)3-s + (−0.826 + 0.563i)5-s + (0.365 − 0.930i)9-s + (−0.623 + 0.781i)11-s + (−0.988 − 0.149i)13-s + (0.365 − 0.930i)15-s + (0.733 + 0.680i)17-s + (0.733 − 0.680i)23-s + (0.365 − 0.930i)25-s + (0.222 + 0.974i)27-s + (−0.955 + 0.294i)29-s − 31-s + (0.0747 − 0.997i)33-s + (0.222 − 0.974i)37-s + (0.900 − 0.433i)39-s + ⋯
L(s)  = 1  + (−0.826 + 0.563i)3-s + (−0.826 + 0.563i)5-s + (0.365 − 0.930i)9-s + (−0.623 + 0.781i)11-s + (−0.988 − 0.149i)13-s + (0.365 − 0.930i)15-s + (0.733 + 0.680i)17-s + (0.733 − 0.680i)23-s + (0.365 − 0.930i)25-s + (0.222 + 0.974i)27-s + (−0.955 + 0.294i)29-s − 31-s + (0.0747 − 0.997i)33-s + (0.222 − 0.974i)37-s + (0.900 − 0.433i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $-0.999 + 0.0124i$
Analytic conductor: \(400.199\)
Root analytic conductor: \(400.199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (1399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (1:\ ),\ -0.999 + 0.0124i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002993739497 + 0.4825904799i\)
\(L(\frac12)\) \(\approx\) \(0.002993739497 + 0.4825904799i\)
\(L(1)\) \(\approx\) \(0.5776876392 + 0.2014138668i\)
\(L(1)\) \(\approx\) \(0.5776876392 + 0.2014138668i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.826 + 0.563i)T \)
5 \( 1 + (-0.826 + 0.563i)T \)
11 \( 1 + (-0.623 + 0.781i)T \)
13 \( 1 + (-0.988 - 0.149i)T \)
17 \( 1 + (0.733 + 0.680i)T \)
23 \( 1 + (0.733 - 0.680i)T \)
29 \( 1 + (-0.955 + 0.294i)T \)
31 \( 1 - T \)
37 \( 1 + (0.222 - 0.974i)T \)
41 \( 1 + (0.826 - 0.563i)T \)
43 \( 1 + (-0.0747 + 0.997i)T \)
47 \( 1 + (-0.988 - 0.149i)T \)
53 \( 1 + (0.733 - 0.680i)T \)
59 \( 1 + (-0.0747 + 0.997i)T \)
61 \( 1 + (-0.955 + 0.294i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.955 + 0.294i)T \)
73 \( 1 + (0.988 - 0.149i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (0.365 - 0.930i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−18.26662546497774308039174021984, −17.24112309108133073026419587629, −16.734136450079326463947554885559, −16.28401655660209944842173137853, −15.5208000009699251734655733828, −14.77497363075447561231293911418, −13.803477298237164448593163404529, −13.100306522960871940619085435620, −12.57241604719503785985352996026, −11.8104378780217072947406522340, −11.37237169346228426553080099458, −10.70040411821249324491941193252, −9.733593422773812096356847766997, −8.99837610269997428149059914763, −7.9194398349790881513804675006, −7.61225220435955922039985600631, −6.9261023446187285318900103466, −5.863279901486739458564195418316, −5.19585907476255499346482506340, −4.76187089626982665747399344201, −3.65384121025988571627675778170, −2.82219486021191753203753503506, −1.73498477350759632901509162179, −0.78755110009858642677790516422, −0.156578821478723169546865233347, 0.6678054550099148480264593026, 1.98000974066932554945555216938, 2.968525195044188234925047517922, 3.7612528439638650418218095586, 4.48205603967067940823706495128, 5.17889865286984684293110470121, 5.88856288576702662074543038501, 6.91505491608673371348444196160, 7.36731660574935160686803527475, 8.09724965630398304440859814173, 9.233419993566818911057997440587, 9.85692316022763812763219893192, 10.684343505639628120875233552235, 10.94588607176292490251485385846, 11.889618439667559418932136746813, 12.546905282742925659975915538342, 12.87337014556883027248879679236, 14.37127448434582598665340763783, 14.97497538560428273470924857743, 15.146615553363709663768959006446, 16.2217309282844546252864403848, 16.598149943997927411059717937260, 17.39410247558879001669814888112, 18.17167462352639524416134738000, 18.56831968517356093737015126782

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