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Properties

Label 1-2100-2100.47-r0-0-0
Degree $1$
Conductor $2100$
Sign $0.999 + 0.0198i$
Analytic cond. $9.75235$
Root an. cond. $9.75235$
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  + (0.913 − 0.406i)11-s + (−0.587 − 0.809i)13-s + (−0.743 + 0.669i)17-s + (0.978 − 0.207i)19-s + (0.994 + 0.104i)23-s + (0.309 + 0.951i)29-s + (0.669 + 0.743i)31-s + (−0.406 + 0.913i)37-s + (−0.809 + 0.587i)41-s − i·43-s + (−0.743 − 0.669i)47-s + (0.207 − 0.978i)53-s + (−0.104 − 0.994i)59-s + (0.104 − 0.994i)61-s + (0.743 − 0.669i)67-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)11-s + (−0.587 − 0.809i)13-s + (−0.743 + 0.669i)17-s + (0.978 − 0.207i)19-s + (0.994 + 0.104i)23-s + (0.309 + 0.951i)29-s + (0.669 + 0.743i)31-s + (−0.406 + 0.913i)37-s + (−0.809 + 0.587i)41-s − i·43-s + (−0.743 − 0.669i)47-s + (0.207 − 0.978i)53-s + (−0.104 − 0.994i)59-s + (0.104 − 0.994i)61-s + (0.743 − 0.669i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0198i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.999 + 0.0198i$
Analytic conductor: \(9.75235\)
Root analytic conductor: \(9.75235\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2100,\ (0:\ ),\ 0.999 + 0.0198i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.699090089 + 0.01687018687i\)
\(L(\frac12)\) \(\approx\) \(1.699090089 + 0.01687018687i\)
\(L(1)\) \(\approx\) \(1.141933032 + 0.01211979118i\)
\(L(1)\) \(\approx\) \(1.141933032 + 0.01211979118i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (-0.587 - 0.809i)T \)
17 \( 1 + (-0.743 + 0.669i)T \)
19 \( 1 + (0.978 - 0.207i)T \)
23 \( 1 + (0.994 + 0.104i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (0.669 + 0.743i)T \)
37 \( 1 + (-0.406 + 0.913i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.743 - 0.669i)T \)
53 \( 1 + (0.207 - 0.978i)T \)
59 \( 1 + (-0.104 - 0.994i)T \)
61 \( 1 + (0.104 - 0.994i)T \)
67 \( 1 + (0.743 - 0.669i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.406 + 0.913i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (0.951 + 0.309i)T \)
89 \( 1 + (0.104 - 0.994i)T \)
97 \( 1 + (0.951 - 0.309i)T \)
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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

−19.73485912053684129275657253456, −19.21679341523772707193019472524, −18.42411596138522245446640252503, −17.55820281360787055951774288418, −17.02142982724091476792030214081, −16.2776331840805894341150288353, −15.41949710152320770660713944111, −14.79392456929414694154151576309, −13.89121384456653383707141175525, −13.49838498813539533902392336877, −12.26225858856396747099892687089, −11.86259508488688445501933348259, −11.12619017745996921043256917394, −10.11861731602328232134509177399, −9.32415257268187520734193038080, −8.9348334570741933922646822782, −7.72530287785432767433942838556, −7.03006579953446183134000756517, −6.42287540409135924853531076360, −5.33244105004135060974866796694, −4.543295938599367601622264879880, −3.83423468319964845600928784816, −2.71394966583916179257340233064, −1.9161527103141486889369812388, −0.794937695340151659464762946225, 0.85233601759782908434707414488, 1.76131440174078403057703322709, 3.03437382704057860770349597981, 3.49734852500107531243037887305, 4.78591798581078271183653904489, 5.24167224035605617036452053742, 6.496176407128125202090687255404, 6.84842563094138911012969057531, 8.01432814100931178373844377298, 8.61116368369843940354323634832, 9.484217016018616193352885080523, 10.185348254750783940282627066288, 11.08986499213027981690783632082, 11.691466744856534261306501538034, 12.56966143102911123375890777955, 13.223993478706213827056335075269, 14.062633982417613129158873313028, 14.76991243170085474920717871634, 15.444253631486938171732471951518, 16.22553600143990423200051271895, 17.08960720919923313716637168459, 17.562387046968235925784511177779, 18.35988158911083851501623293688, 19.26066707954813901993706244412, 19.86127084725052381683290223746

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