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Properties

Label 1-3724-3724.1327-r1-0-0
Degree $1$
Conductor $3724$
Sign $-0.0738 - 0.997i$
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  + (−0.698 − 0.715i)3-s + (−0.969 + 0.246i)5-s + (−0.0249 + 0.999i)9-s + (−0.365 − 0.930i)11-s + (0.980 − 0.198i)13-s + (0.853 + 0.521i)15-s + (0.456 + 0.889i)17-s + (0.998 − 0.0498i)23-s + (0.878 − 0.478i)25-s + (0.733 − 0.680i)27-s + (0.921 + 0.388i)29-s − 31-s + (−0.411 + 0.911i)33-s + (−0.733 − 0.680i)37-s + (−0.826 − 0.563i)39-s + ⋯
L(s)  = 1  + (−0.698 − 0.715i)3-s + (−0.969 + 0.246i)5-s + (−0.0249 + 0.999i)9-s + (−0.365 − 0.930i)11-s + (0.980 − 0.198i)13-s + (0.853 + 0.521i)15-s + (0.456 + 0.889i)17-s + (0.998 − 0.0498i)23-s + (0.878 − 0.478i)25-s + (0.733 − 0.680i)27-s + (0.921 + 0.388i)29-s − 31-s + (−0.411 + 0.911i)33-s + (−0.733 − 0.680i)37-s + (−0.826 − 0.563i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0738 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0738 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $-0.0738 - 0.997i$
Analytic conductor: \(400.199\)
Root analytic conductor: \(400.199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (1327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (1:\ ),\ -0.0738 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7493217768 - 0.8068439916i\)
\(L(\frac12)\) \(\approx\) \(0.7493217768 - 0.8068439916i\)
\(L(1)\) \(\approx\) \(0.7147701752 - 0.1750716936i\)
\(L(1)\) \(\approx\) \(0.7147701752 - 0.1750716936i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.698 - 0.715i)T \)
5 \( 1 + (-0.969 + 0.246i)T \)
11 \( 1 + (-0.365 - 0.930i)T \)
13 \( 1 + (0.980 - 0.198i)T \)
17 \( 1 + (0.456 + 0.889i)T \)
23 \( 1 + (0.998 - 0.0498i)T \)
29 \( 1 + (0.921 + 0.388i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.733 - 0.680i)T \)
41 \( 1 + (-0.969 + 0.246i)T \)
43 \( 1 + (-0.995 + 0.0995i)T \)
47 \( 1 + (-0.980 + 0.198i)T \)
53 \( 1 + (0.456 - 0.889i)T \)
59 \( 1 + (0.411 - 0.911i)T \)
61 \( 1 + (0.921 + 0.388i)T \)
67 \( 1 + (0.939 - 0.342i)T \)
71 \( 1 + (0.124 - 0.992i)T \)
73 \( 1 + (-0.318 - 0.947i)T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (0.988 - 0.149i)T \)
89 \( 1 + (-0.853 - 0.521i)T \)
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

−18.48062819657381832335782765879, −18.023323054911292789251445582357, −17.06265849109065108999255541554, −16.54423058352846013193613756532, −15.84774552667403629920950136426, −15.388242949739182520484871872433, −14.83294378433683809003355368021, −13.85476515055016448357066765575, −12.913090207094967606833562282393, −12.29824513921686479833177622053, −11.55534378674100223832220967845, −11.199608274719808918540850473266, −10.26723249930929339821942606648, −9.72555515586015087199387825051, −8.78176759564312933602975486719, −8.262289676377270471301530619377, −7.04313186627729424697000040899, −6.85042162923255585999711943216, −5.53607567149586851711694961529, −5.02595962005861900406128123484, −4.33173199065366465794260925259, −3.59949448188059387460988074411, −2.89067203841383431960613539515, −1.46079577424715249995827851079, −0.61567942549651119755189915588, 0.33361159192288927189724924881, 1.04224589136429052973025296623, 1.9763743391514858798383885301, 3.27697192987231916382496057481, 3.56517230115621698162570080230, 4.85046950281378282516128094770, 5.429589916288566734463575760665, 6.37193428072281681427671596203, 6.80858831365570702612582940021, 7.77274818685490601435550022170, 8.28458791060064929130294395191, 8.8486397722075511099191342335, 10.34702821072169687049195322649, 10.73426788126893789151909918893, 11.3972492061961200516177344355, 11.92477764240404708019520173250, 12.87622296775702452130489407225, 13.15974454210080104422028457993, 14.13396706508407060275270049076, 14.853596775985125876706435327018, 15.72094716537670219282037242272, 16.30942349562009293043942717233, 16.771712301473413777075102754088, 17.7581415521026802820214583947, 18.32250785370886995623526856495

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