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Properties

Label 1-1573-1573.1025-r0-0-0
Degree $1$
Conductor $1573$
Sign $-0.367 - 0.930i$
Analytic cond. $7.30498$
Root an. cond. $7.30498$
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  + (0.647 + 0.761i)2-s + (0.669 + 0.743i)3-s + (−0.161 + 0.986i)4-s + (−0.226 + 0.974i)5-s + (−0.132 + 0.991i)6-s + (−0.299 + 0.953i)7-s + (−0.856 + 0.516i)8-s + (−0.104 + 0.994i)9-s + (−0.888 + 0.458i)10-s + (−0.841 + 0.540i)12-s + (−0.921 + 0.389i)14-s + (−0.875 + 0.483i)15-s + (−0.948 − 0.318i)16-s + (−0.532 + 0.846i)17-s + (−0.825 + 0.564i)18-s + (0.938 − 0.345i)19-s + ⋯
L(s)  = 1  + (0.647 + 0.761i)2-s + (0.669 + 0.743i)3-s + (−0.161 + 0.986i)4-s + (−0.226 + 0.974i)5-s + (−0.132 + 0.991i)6-s + (−0.299 + 0.953i)7-s + (−0.856 + 0.516i)8-s + (−0.104 + 0.994i)9-s + (−0.888 + 0.458i)10-s + (−0.841 + 0.540i)12-s + (−0.921 + 0.389i)14-s + (−0.875 + 0.483i)15-s + (−0.948 − 0.318i)16-s + (−0.532 + 0.846i)17-s + (−0.825 + 0.564i)18-s + (0.938 − 0.345i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.367 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.367 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1573\)    =    \(11^{2} \cdot 13\)
Sign: $-0.367 - 0.930i$
Analytic conductor: \(7.30498\)
Root analytic conductor: \(7.30498\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1573} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1573,\ (0:\ ),\ -0.367 - 0.930i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.156864723 + 1.700348155i\)
\(L(\frac12)\) \(\approx\) \(-1.156864723 + 1.700348155i\)
\(L(1)\) \(\approx\) \(0.5985023710 + 1.390465265i\)
\(L(1)\) \(\approx\) \(0.5985023710 + 1.390465265i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.647 + 0.761i)T \)
3 \( 1 + (0.669 + 0.743i)T \)
5 \( 1 + (-0.226 + 0.974i)T \)
7 \( 1 + (-0.299 + 0.953i)T \)
17 \( 1 + (-0.532 + 0.846i)T \)
19 \( 1 + (0.938 - 0.345i)T \)
23 \( 1 + (-0.580 - 0.814i)T \)
29 \( 1 + (0.0665 - 0.997i)T \)
31 \( 1 + (0.931 - 0.362i)T \)
37 \( 1 + (-0.703 + 0.710i)T \)
41 \( 1 + (-0.524 + 0.851i)T \)
43 \( 1 + (0.981 + 0.189i)T \)
47 \( 1 + (0.825 + 0.564i)T \)
53 \( 1 + (0.198 - 0.980i)T \)
59 \( 1 + (0.524 + 0.851i)T \)
61 \( 1 + (0.761 + 0.647i)T \)
67 \( 1 + (0.971 + 0.235i)T \)
71 \( 1 + (0.962 + 0.272i)T \)
73 \( 1 + (-0.113 + 0.993i)T \)
79 \( 1 + (-0.0855 - 0.996i)T \)
83 \( 1 + (-0.491 + 0.870i)T \)
89 \( 1 + (-0.618 + 0.786i)T \)
97 \( 1 + (0.730 - 0.683i)T \)
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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

−20.116469739997144269371705550437, −19.58111227070638081653620370504, −18.802927573906156974347622225699, −17.91637124553325972709375555098, −17.19305077181276950380249694074, −15.92494892080678816234186275957, −15.58343094811793768187178208699, −14.11001333862586832720966236607, −13.95181420120644117600643903312, −13.23659179198349207856787376179, −12.439512113834757788188640411644, −11.97175435238916841768231112346, −11.05610325634574575558806873226, −9.9640806819258099856675047621, −9.33182156508265170382046354260, −8.57134561194887838984577900268, −7.49048858595913230490189839312, −6.86889770683103751446655750777, −5.72775775763681894899362103893, −4.88845977733351961284678171988, −3.87380425959003545298021700131, −3.38097239138749260664457612108, −2.22341747154303772424937389230, −1.29815447702741095428672561076, −0.55432707139862726016142892903, 2.368298469295011421882796813463, 2.74442132346798135823941433183, 3.71296166070520806783656996242, 4.37301633145805863844815126818, 5.41850132558984979927701830448, 6.2028159395404644246149060598, 6.96384091161624003533714923977, 8.05540444992368418329654818143, 8.45260966754974728793556636282, 9.48421175495379538572062262111, 10.2041566369480950795426824731, 11.31280455883900757946462787475, 11.93141231188220246162487152595, 12.99272508732001738181392314116, 13.76693113969347678995638779316, 14.43936353329121559204732591544, 15.13323271118651754889418209917, 15.58932995718236960434738800361, 16.097026128888859687993571988885, 17.165587807090951161571046466772, 17.99031504547917685662591731503, 18.830406118381151015872634693627, 19.49855965907632058315566025109, 20.495549685685118280328200345017, 21.263701316566682715540963534408

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