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Properties

Label 1-2001-2001.1139-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.211 + 0.977i$
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  + (−0.891 + 0.452i)2-s + (0.591 − 0.806i)4-s + (0.101 − 0.994i)5-s + (−0.933 − 0.359i)7-s + (−0.162 + 0.986i)8-s + (0.359 + 0.933i)10-s + (−0.639 + 0.768i)11-s + (−0.557 − 0.830i)13-s + (0.994 − 0.101i)14-s + (−0.301 − 0.953i)16-s + (0.755 + 0.654i)17-s + (−0.806 − 0.591i)19-s + (−0.742 − 0.670i)20-s + (0.222 − 0.974i)22-s + (−0.979 − 0.202i)25-s + (0.872 + 0.488i)26-s + ⋯
L(s)  = 1  + (−0.891 + 0.452i)2-s + (0.591 − 0.806i)4-s + (0.101 − 0.994i)5-s + (−0.933 − 0.359i)7-s + (−0.162 + 0.986i)8-s + (0.359 + 0.933i)10-s + (−0.639 + 0.768i)11-s + (−0.557 − 0.830i)13-s + (0.994 − 0.101i)14-s + (−0.301 − 0.953i)16-s + (0.755 + 0.654i)17-s + (−0.806 − 0.591i)19-s + (−0.742 − 0.670i)20-s + (0.222 − 0.974i)22-s + (−0.979 − 0.202i)25-s + (0.872 + 0.488i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.211 + 0.977i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.211 + 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2809816671 + 0.2267182009i\)
\(L(\frac12)\) \(\approx\) \(0.2809816671 + 0.2267182009i\)
\(L(1)\) \(\approx\) \(0.5318383330 + 0.007788132287i\)
\(L(1)\) \(\approx\) \(0.5318383330 + 0.007788132287i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.891 + 0.452i)T \)
5 \( 1 + (0.101 - 0.994i)T \)
7 \( 1 + (-0.933 - 0.359i)T \)
11 \( 1 + (-0.639 + 0.768i)T \)
13 \( 1 + (-0.557 - 0.830i)T \)
17 \( 1 + (0.755 + 0.654i)T \)
19 \( 1 + (-0.806 - 0.591i)T \)
31 \( 1 + (-0.925 - 0.377i)T \)
37 \( 1 + (-0.852 + 0.523i)T \)
41 \( 1 + (0.540 + 0.841i)T \)
43 \( 1 + (0.925 - 0.377i)T \)
47 \( 1 + (-0.433 - 0.900i)T \)
53 \( 1 + (-0.685 - 0.728i)T \)
59 \( 1 + (0.142 + 0.989i)T \)
61 \( 1 + (0.998 - 0.0611i)T \)
67 \( 1 + (0.768 - 0.639i)T \)
71 \( 1 + (0.0203 + 0.999i)T \)
73 \( 1 + (-0.983 - 0.182i)T \)
79 \( 1 + (-0.953 - 0.301i)T \)
83 \( 1 + (-0.339 + 0.940i)T \)
89 \( 1 + (0.574 + 0.818i)T \)
97 \( 1 + (-0.699 + 0.714i)T \)
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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

−19.33631017077436549648151695604, −19.013434940073581330759144311238, −18.66561113961174406823794753923, −17.75702401864031523626596878592, −16.98168025077492533016397355793, −16.06428275767874843646877267622, −15.86142704168346434056977043010, −14.62078076409291163083111829472, −14.016411539958814597336421402439, −12.91170247964046528101405958856, −12.34144648567832618924359843049, −11.46274832140293904993778115488, −10.77877847826672851081250626833, −10.115162016510890693262242650762, −9.44798732724662335101043646154, −8.74353553333543926112830785726, −7.69786565098641299584088487132, −7.10531560675498128914112193314, −6.31398974683380990455744928568, −5.5694265676010107133610104093, −4.02017021723043358394209738068, −3.20518897413162180391132780612, −2.63050346765690675322073687327, −1.790845757398483357561186311917, −0.22061908085339670948769952356, 0.76932789392556120151530346166, 1.87401558419480489351410416410, 2.78455914356666080988544995733, 4.03376875465000087611921472410, 5.10264921461786853768802741230, 5.66197485063878203222808696344, 6.62608332312117384734772437517, 7.43801863655528023362767384279, 8.10043355593648557440940306949, 8.87576510993112547212831352223, 9.765705251522986989852884162727, 10.10094565198252342808284593967, 10.92188105909518758114153256115, 12.12413029505577005486291107085, 12.786198568244479747109639121838, 13.29909097188089263806315484251, 14.532530820212584639325019216633, 15.2251329381118422316211172330, 15.90298647382789849093594655512, 16.54817504052123660071377422230, 17.26209505891449946057770926416, 17.633125985174960256308700100414, 18.68147413014100952422807815559, 19.43018333552554658482242656231, 19.96308603360959931714518027291

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