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Properties

Label 2-2000-100.31-c0-0-0
Degree $2$
Conductor $2000$
Sign $0.999 + 0.0119i$
Analytic cond. $0.998130$
Root an. cond. $0.999064$
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  + (0.309 − 0.951i)9-s + (−0.363 + 1.11i)13-s + (1.53 + 1.11i)17-s + (0.5 − 0.363i)29-s + (0.587 − 1.80i)37-s + (0.5 − 1.53i)41-s + 49-s + (−0.951 + 0.690i)53-s + (0.5 + 1.53i)61-s + (0.363 + 1.11i)73-s + (−0.809 − 0.587i)81-s + (−0.190 − 0.587i)89-s + (−1.53 + 1.11i)97-s + 0.618·101-s + (−0.5 + 1.53i)109-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)9-s + (−0.363 + 1.11i)13-s + (1.53 + 1.11i)17-s + (0.5 − 0.363i)29-s + (0.587 − 1.80i)37-s + (0.5 − 1.53i)41-s + 49-s + (−0.951 + 0.690i)53-s + (0.5 + 1.53i)61-s + (0.363 + 1.11i)73-s + (−0.809 − 0.587i)81-s + (−0.190 − 0.587i)89-s + (−1.53 + 1.11i)97-s + 0.618·101-s + (−0.5 + 1.53i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0119i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0119i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2000\)    =    \(2^{4} \cdot 5^{3}\)
Sign: $0.999 + 0.0119i$
Analytic conductor: \(0.998130\)
Root analytic conductor: \(0.999064\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2000} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2000,\ (\ :0),\ 0.999 + 0.0119i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.226677684\)
\(L(\frac12)\) \(\approx\) \(1.226677684\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.309 + 0.951i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.587 + 1.80i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.309 - 0.951i)T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.331635224012857765542338454093, −8.704885312870159891691917423911, −7.68317823301116862650968337570, −7.05346314941187547529294262395, −6.14029733976521658349422507521, −5.50812089311200934233797220077, −4.19422591319230789415155372944, −3.74935178407869263455676998620, −2.44942633265448784985370316401, −1.20726570994757818288573132535, 1.18301575992665757179305782362, 2.63253375070817346907288119099, 3.32792880088743525522245812877, 4.75176391173091011651157002334, 5.15999341484507627187051428567, 6.13104954175784424368958083218, 7.17166073111331420703966369496, 7.87550797230224740235584754898, 8.293039712849236862453808445113, 9.653247330233386310778488147322

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