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Properties

Label 1-2000-2000.691-r1-0-0
Degree $1$
Conductor $2000$
Sign $0.880 + 0.473i$
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  + (−0.684 − 0.728i)3-s + (−0.809 + 0.587i)7-s + (−0.0627 + 0.998i)9-s + (0.844 + 0.535i)11-s + (0.998 + 0.0627i)13-s + (−0.425 + 0.904i)17-s + (0.684 − 0.728i)19-s + (0.982 + 0.187i)21-s + (0.968 + 0.248i)23-s + (0.770 − 0.637i)27-s + (−0.481 + 0.876i)29-s + (0.425 − 0.904i)31-s + (−0.187 − 0.982i)33-s + (0.770 + 0.637i)37-s + (−0.637 − 0.770i)39-s + ⋯
L(s)  = 1  + (−0.684 − 0.728i)3-s + (−0.809 + 0.587i)7-s + (−0.0627 + 0.998i)9-s + (0.844 + 0.535i)11-s + (0.998 + 0.0627i)13-s + (−0.425 + 0.904i)17-s + (0.684 − 0.728i)19-s + (0.982 + 0.187i)21-s + (0.968 + 0.248i)23-s + (0.770 − 0.637i)27-s + (−0.481 + 0.876i)29-s + (0.425 − 0.904i)31-s + (−0.187 − 0.982i)33-s + (0.770 + 0.637i)37-s + (−0.637 − 0.770i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2000\)    =    \(2^{4} \cdot 5^{3}\)
Sign: $0.880 + 0.473i$
Analytic conductor: \(214.929\)
Root analytic conductor: \(214.929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2000} (691, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2000,\ (1:\ ),\ 0.880 + 0.473i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.763877088 + 0.4440372642i\)
\(L(\frac12)\) \(\approx\) \(1.763877088 + 0.4440372642i\)
\(L(1)\) \(\approx\) \(0.9404630200 + 0.01616325697i\)
\(L(1)\) \(\approx\) \(0.9404630200 + 0.01616325697i\)

Euler product

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

−19.82704564764504359446253295176, −18.918009514283131589809650915542, −18.24623234313336579451568506046, −17.31250273346475388660668773089, −16.742112767362800794079635190985, −16.080265501485021086699501931226, −15.66116562264718282847594899499, −14.61348599286026425051709331549, −13.81594641018641182688350785719, −13.17294239395371650721585629458, −12.13096400736097941444269409857, −11.519487356099055890620757325890, −10.77158248675077174385204482335, −10.14514296892438544041810755607, −9.22644390184283034724701805964, −8.84191335914479462838206225441, −7.47364754843922204567704307181, −6.648280028250580662964609550907, −6.04407683143396835642650346248, −5.25830208123429638574124438890, −4.14062485239768846744114808863, −3.6815673959987162376764307380, −2.8218868554256015458783821438, −1.12502015852479618094115423966, −0.55565964410120708838994879622, 0.75685727073263368351231426400, 1.55998484657780839770401828698, 2.55316970986076722010741264299, 3.554445008586017595840990369890, 4.5587889358493291648405915237, 5.55645941287644199801302987976, 6.25423415707480848045662091602, 6.78763325871581140575462747698, 7.5901794777622847319812650953, 8.71011667367946580633105309319, 9.230998679777309525333708449598, 10.25583031721301588348869654095, 11.14268478145942853545676587021, 11.71097054556809237157546224463, 12.47984639508351838542597955084, 13.18757418905612012962992154052, 13.60785174558542094135174553723, 14.849447998442788896097025682724, 15.47016094602156189352467326577, 16.32828048627802983055239846124, 16.98383724777298619027724489127, 17.65883362833332356464731701723, 18.43696075015635234096028248217, 18.97500308336619469539367080878, 19.70714412767712249132713821966

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