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Properties

Label 1-2001-2001.1217-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.874 - 0.484i$
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  + (0.415 + 0.909i)2-s + (−0.654 + 0.755i)4-s + (0.841 + 0.540i)5-s + (0.142 + 0.989i)7-s + (−0.959 − 0.281i)8-s + (−0.142 + 0.989i)10-s + (−0.415 + 0.909i)11-s + (−0.142 + 0.989i)13-s + (−0.841 + 0.540i)14-s + (−0.142 − 0.989i)16-s + (0.654 + 0.755i)17-s + (−0.654 + 0.755i)19-s + (−0.959 + 0.281i)20-s − 22-s + (0.415 + 0.909i)25-s + (−0.959 + 0.281i)26-s + ⋯
L(s)  = 1  + (0.415 + 0.909i)2-s + (−0.654 + 0.755i)4-s + (0.841 + 0.540i)5-s + (0.142 + 0.989i)7-s + (−0.959 − 0.281i)8-s + (−0.142 + 0.989i)10-s + (−0.415 + 0.909i)11-s + (−0.142 + 0.989i)13-s + (−0.841 + 0.540i)14-s + (−0.142 − 0.989i)16-s + (0.654 + 0.755i)17-s + (−0.654 + 0.755i)19-s + (−0.959 + 0.281i)20-s − 22-s + (0.415 + 0.909i)25-s + (−0.959 + 0.281i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.874 - 0.484i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.874 - 0.484i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4994657035 + 1.933246061i\)
\(L(\frac12)\) \(\approx\) \(-0.4994657035 + 1.933246061i\)
\(L(1)\) \(\approx\) \(0.7611682575 + 1.126113428i\)
\(L(1)\) \(\approx\) \(0.7611682575 + 1.126113428i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.415 + 0.909i)T \)
5 \( 1 + (0.841 + 0.540i)T \)
7 \( 1 + (0.142 + 0.989i)T \)
11 \( 1 + (-0.415 + 0.909i)T \)
13 \( 1 + (-0.142 + 0.989i)T \)
17 \( 1 + (0.654 + 0.755i)T \)
19 \( 1 + (-0.654 + 0.755i)T \)
31 \( 1 + (0.959 + 0.281i)T \)
37 \( 1 + (0.841 - 0.540i)T \)
41 \( 1 + (0.841 + 0.540i)T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.142 - 0.989i)T \)
59 \( 1 + (0.142 - 0.989i)T \)
61 \( 1 + (-0.959 - 0.281i)T \)
67 \( 1 + (-0.415 - 0.909i)T \)
71 \( 1 + (-0.415 - 0.909i)T \)
73 \( 1 + (0.654 - 0.755i)T \)
79 \( 1 + (-0.142 + 0.989i)T \)
83 \( 1 + (0.841 - 0.540i)T \)
89 \( 1 + (0.959 - 0.281i)T \)
97 \( 1 + (0.841 + 0.540i)T \)
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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

−19.76828242304377649557300482115, −18.90146782805281091397836864811, −18.14407500599868717489340296953, −17.410683099104297714495944251289, −16.83105436647510501740804008087, −15.86914281994694293562855247218, −14.91695874812077727888524624511, −14.03715202280547499658254390102, −13.49345877758390206550113789207, −13.104246290795251233630751999195, −12.21956599934003386569309557807, −11.334043967349571180428917177087, −10.48639814758866566936812208391, −10.162690938333444691983437295470, −9.23252411442544133877737205001, −8.46351692171971870906813338263, −7.559012140474292102373619441647, −6.31446547790508368369795246476, −5.60601243991053173213945869510, −4.883613224306338452892105803351, −4.14588586930521537325942565624, −3.01960772007429936949655987766, −2.4782402502675105675156883766, −1.13189519376286259493015590397, −0.64356296638909189848615763818, 1.74907438025669073315498854495, 2.444092849407800469115709342901, 3.44991087357898773991272977287, 4.529508811165598116894054292160, 5.20771776367559790013228426311, 6.192566489040776953424151655921, 6.40563755869683522695843948952, 7.543194929453007929906436361270, 8.20965735481650145819109881425, 9.194722131789495668956955868710, 9.710391127667041548403683139033, 10.621835205279281123944689706410, 11.8019013211316960232508336043, 12.45143554201502189604396256015, 13.08075421976385639055708370360, 14.00956832848782131551381240262, 14.716057544599361275032113935463, 14.98476580727856734123805951374, 15.93162736491434100871206658528, 16.76700236065411781613830819167, 17.382192438558400411227860381986, 18.14358943451042118772616369500, 18.6188144274435991670580457029, 19.37871640324842797610586321069, 20.824186320133476359258282763660

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