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Properties

Label 1-2000-2000.1011-r1-0-0
Degree $1$
Conductor $2000$
Sign $-0.605 - 0.795i$
Analytic cond. $214.929$
Root an. cond. $214.929$
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  + (0.904 + 0.425i)3-s + (−0.809 − 0.587i)7-s + (0.637 + 0.770i)9-s + (0.998 + 0.0627i)11-s + (0.770 − 0.637i)13-s + (−0.992 + 0.125i)17-s + (−0.904 + 0.425i)19-s + (−0.481 − 0.876i)21-s + (−0.929 − 0.368i)23-s + (0.248 + 0.968i)27-s + (−0.684 + 0.728i)29-s + (0.992 − 0.125i)31-s + (0.876 + 0.481i)33-s + (0.248 − 0.968i)37-s + (0.968 − 0.248i)39-s + ⋯
L(s)  = 1  + (0.904 + 0.425i)3-s + (−0.809 − 0.587i)7-s + (0.637 + 0.770i)9-s + (0.998 + 0.0627i)11-s + (0.770 − 0.637i)13-s + (−0.992 + 0.125i)17-s + (−0.904 + 0.425i)19-s + (−0.481 − 0.876i)21-s + (−0.929 − 0.368i)23-s + (0.248 + 0.968i)27-s + (−0.684 + 0.728i)29-s + (0.992 − 0.125i)31-s + (0.876 + 0.481i)33-s + (0.248 − 0.968i)37-s + (0.968 − 0.248i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2000\)    =    \(2^{4} \cdot 5^{3}\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(214.929\)
Root analytic conductor: \(214.929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2000} (1011, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2000,\ (1:\ ),\ -0.605 - 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4491525815 - 0.9060028204i\)
\(L(\frac12)\) \(\approx\) \(0.4491525815 - 0.9060028204i\)
\(L(1)\) \(\approx\) \(1.204364876 + 0.02564148145i\)
\(L(1)\) \(\approx\) \(1.204364876 + 0.02564148145i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.904 + 0.425i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (0.998 + 0.0627i)T \)
13 \( 1 + (0.770 - 0.637i)T \)
17 \( 1 + (-0.992 + 0.125i)T \)
19 \( 1 + (-0.904 + 0.425i)T \)
23 \( 1 + (-0.929 - 0.368i)T \)
29 \( 1 + (-0.684 + 0.728i)T \)
31 \( 1 + (0.992 - 0.125i)T \)
37 \( 1 + (0.248 - 0.968i)T \)
41 \( 1 + (0.929 - 0.368i)T \)
43 \( 1 + (-0.951 + 0.309i)T \)
47 \( 1 + (0.187 + 0.982i)T \)
53 \( 1 + (-0.481 - 0.876i)T \)
59 \( 1 + (0.844 + 0.535i)T \)
61 \( 1 + (0.368 - 0.929i)T \)
67 \( 1 + (-0.684 - 0.728i)T \)
71 \( 1 + (-0.187 - 0.982i)T \)
73 \( 1 + (-0.535 - 0.844i)T \)
79 \( 1 + (0.425 - 0.904i)T \)
83 \( 1 + (-0.904 + 0.425i)T \)
89 \( 1 + (-0.535 - 0.844i)T \)
97 \( 1 + (0.728 + 0.684i)T \)
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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

−19.89830541878158951562545984745, −19.27407853386080409103588883555, −18.80872499194872088767741374777, −18.00354505415149067315757446971, −17.18690099060685506208182514600, −16.25169229374850300576981907082, −15.496443109659048110555765984730, −14.995335557770114981605147649805, −14.03662577261145860066489747553, −13.42652327048715251384215949589, −12.884693896546302254983358361313, −11.8851106878936165871236348486, −11.41594282516629569008305253015, −10.081754713158293341826199206016, −9.43726764921496541230293681493, −8.728576868389928616679273827146, −8.30446698463875718741152514641, −6.9961764823712723618853743843, −6.53310247134225698498616265192, −5.878949825835136356308712307444, −4.28845993145904609966448853339, −3.88371618402736893503782593796, −2.79669822787539867163762311667, −2.10305848158949484563890177956, −1.164822788840282993595563991329, 0.145575774762309695717525755643, 1.414674752836712890163384493104, 2.3489308036447295271804165164, 3.39342955579702817741386370214, 3.94829729404819092092259138074, 4.586732783691612686856428631682, 6.017902434909688929934538989184, 6.57669390511418574971731085964, 7.54553089647827281614919504491, 8.37372693379580559589875309689, 9.04418657733907060792208837722, 9.74085522940327821580959381892, 10.53258741523955055893709241861, 11.08571281880268448169766029407, 12.36685765494440730220286758395, 13.04410042905562781066563182446, 13.649158889215778843696485355485, 14.43383524245954214190608599675, 15.04478783302223568008828183791, 15.969555617622946954955630274251, 16.34348874454623261409287467753, 17.28664438328262678316563567616, 18.0830242759280331662032837620, 19.11636320754175845399868686344, 19.5671902559146566832711290635

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