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Properties

Label 1-2001-2001.497-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.694 - 0.719i$
Analytic cond. $9.29260$
Root an. cond. $9.29260$
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  + (−0.992 + 0.122i)2-s + (0.970 − 0.242i)4-s + (0.986 + 0.162i)5-s + (0.999 − 0.0407i)7-s + (−0.933 + 0.359i)8-s + (−0.999 − 0.0407i)10-s + (0.714 − 0.699i)11-s + (−0.591 − 0.806i)13-s + (−0.986 + 0.162i)14-s + (0.882 − 0.470i)16-s + (−0.415 − 0.909i)17-s + (0.970 − 0.242i)19-s + (0.996 − 0.0815i)20-s + (−0.623 + 0.781i)22-s + (0.947 + 0.320i)25-s + (0.685 + 0.728i)26-s + ⋯
L(s)  = 1  + (−0.992 + 0.122i)2-s + (0.970 − 0.242i)4-s + (0.986 + 0.162i)5-s + (0.999 − 0.0407i)7-s + (−0.933 + 0.359i)8-s + (−0.999 − 0.0407i)10-s + (0.714 − 0.699i)11-s + (−0.591 − 0.806i)13-s + (−0.986 + 0.162i)14-s + (0.882 − 0.470i)16-s + (−0.415 − 0.909i)17-s + (0.970 − 0.242i)19-s + (0.996 − 0.0815i)20-s + (−0.623 + 0.781i)22-s + (0.947 + 0.320i)25-s + (0.685 + 0.728i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.694 - 0.719i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.694 - 0.719i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.389404342 - 0.5898447577i\)
\(L(\frac12)\) \(\approx\) \(1.389404342 - 0.5898447577i\)
\(L(1)\) \(\approx\) \(0.9827853556 - 0.1153887508i\)
\(L(1)\) \(\approx\) \(0.9827853556 - 0.1153887508i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.992 + 0.122i)T \)
5 \( 1 + (0.986 + 0.162i)T \)
7 \( 1 + (0.999 - 0.0407i)T \)
11 \( 1 + (0.714 - 0.699i)T \)
13 \( 1 + (-0.591 - 0.806i)T \)
17 \( 1 + (-0.415 - 0.909i)T \)
19 \( 1 + (0.970 - 0.242i)T \)
31 \( 1 + (0.301 - 0.953i)T \)
37 \( 1 + (-0.0611 + 0.998i)T \)
41 \( 1 + (-0.959 + 0.281i)T \)
43 \( 1 + (-0.301 - 0.953i)T \)
47 \( 1 + (-0.222 + 0.974i)T \)
53 \( 1 + (0.262 + 0.965i)T \)
59 \( 1 + (0.654 - 0.755i)T \)
61 \( 1 + (-0.862 - 0.505i)T \)
67 \( 1 + (0.714 + 0.699i)T \)
71 \( 1 + (-0.339 - 0.940i)T \)
73 \( 1 + (-0.0203 - 0.999i)T \)
79 \( 1 + (0.882 + 0.470i)T \)
83 \( 1 + (-0.377 - 0.925i)T \)
89 \( 1 + (-0.557 + 0.830i)T \)
97 \( 1 + (-0.818 - 0.574i)T \)
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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

−19.96455332807155304558176443331, −19.38713556377344074623767287983, −18.26252232407105940912517538859, −17.92988995923455845318767455504, −17.18417010069088572156986571413, −16.80902703426521229571146722425, −15.85667984547912576979130061951, −14.77564753052968012824967254806, −14.46282509534146286908736811544, −13.463960342113314730590534850984, −12.3723538125202039974392517924, −11.867753112390276640268059993301, −11.041557361886170629502878365178, −10.191081091451083993903253148034, −9.62358835420105734079202050209, −8.87110030713072585041612981163, −8.26152703709897301601034456116, −7.17135449477142028897100329769, −6.69288758479760607668065986192, −5.6662920798483886946819630780, −4.830692937094838312061766958854, −3.783914297025748709461813733650, −2.465214104032986616261399946672, −1.76290135002460499365577538160, −1.26748965384787176080552842054, 0.75706524292230297280484717764, 1.571104087027065001635353895451, 2.49991472217309936430308074795, 3.23661412923815334586893991695, 4.82190923762402182103907903182, 5.48759970563096674390901698230, 6.312598790597882523803312945074, 7.13557979802889957152067923398, 7.88912489129275119012482847333, 8.69687654358384621087712876710, 9.43266757759391806499468657573, 10.02689682632570391109744867457, 10.896103222779053743971433514983, 11.50467835748636708257191106580, 12.18792922637503880239912526319, 13.51011102431279979787337407751, 14.02253671421157800578949503160, 14.86490976794421788228496031406, 15.48246212918318736731724615798, 16.51918436515383067509040915924, 17.17064420309764755416355211015, 17.62013012659112259656632178334, 18.334551688523241806223741318564, 18.84363031267270375834494699847, 19.974621524816312736689538232041

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