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Properties

Label 2-4000-125.58-c0-0-0
Degree $2$
Conductor $4000$
Sign $0.266 - 0.963i$
Analytic cond. $1.99626$
Root an. cond. $1.41289$
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  + (0.481 + 0.876i)5-s + (0.844 − 0.535i)9-s + (−0.344 + 1.54i)13-s + (−1.95 + 0.0613i)17-s + (−0.535 + 0.844i)25-s + (1.96 + 0.374i)29-s + (1.31 + 1.49i)37-s + (−0.872 + 1.05i)41-s + (0.876 + 0.481i)45-s + (0.951 + 0.309i)49-s + (−0.888 − 0.689i)53-s + (−0.614 − 0.742i)61-s + (−1.51 + 0.440i)65-s + (−0.957 − 1.61i)73-s + (0.425 − 0.904i)81-s + ⋯
L(s)  = 1  + (0.481 + 0.876i)5-s + (0.844 − 0.535i)9-s + (−0.344 + 1.54i)13-s + (−1.95 + 0.0613i)17-s + (−0.535 + 0.844i)25-s + (1.96 + 0.374i)29-s + (1.31 + 1.49i)37-s + (−0.872 + 1.05i)41-s + (0.876 + 0.481i)45-s + (0.951 + 0.309i)49-s + (−0.888 − 0.689i)53-s + (−0.614 − 0.742i)61-s + (−1.51 + 0.440i)65-s + (−0.957 − 1.61i)73-s + (0.425 − 0.904i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.266 - 0.963i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (2433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.266 - 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.354109936\)
\(L(\frac12)\) \(\approx\) \(1.354109936\)
\(L(1)\) 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 \( 1 + (-0.481 - 0.876i)T \)
good3 \( 1 + (-0.844 + 0.535i)T^{2} \)
7 \( 1 + (-0.951 - 0.309i)T^{2} \)
11 \( 1 + (0.728 - 0.684i)T^{2} \)
13 \( 1 + (0.344 - 1.54i)T + (-0.904 - 0.425i)T^{2} \)
17 \( 1 + (1.95 - 0.0613i)T + (0.998 - 0.0627i)T^{2} \)
19 \( 1 + (-0.535 + 0.844i)T^{2} \)
23 \( 1 + (-0.982 - 0.187i)T^{2} \)
29 \( 1 + (-1.96 - 0.374i)T + (0.929 + 0.368i)T^{2} \)
31 \( 1 + (0.0627 + 0.998i)T^{2} \)
37 \( 1 + (-1.31 - 1.49i)T + (-0.125 + 0.992i)T^{2} \)
41 \( 1 + (0.872 - 1.05i)T + (-0.187 - 0.982i)T^{2} \)
43 \( 1 + (0.587 - 0.809i)T^{2} \)
47 \( 1 + (0.770 + 0.637i)T^{2} \)
53 \( 1 + (0.888 + 0.689i)T + (0.248 + 0.968i)T^{2} \)
59 \( 1 + (-0.876 + 0.481i)T^{2} \)
61 \( 1 + (0.614 + 0.742i)T + (-0.187 + 0.982i)T^{2} \)
67 \( 1 + (0.368 + 0.929i)T^{2} \)
71 \( 1 + (-0.637 + 0.770i)T^{2} \)
73 \( 1 + (0.957 + 1.61i)T + (-0.481 + 0.876i)T^{2} \)
79 \( 1 + (-0.535 - 0.844i)T^{2} \)
83 \( 1 + (0.844 + 0.535i)T^{2} \)
89 \( 1 + (0.183 - 0.713i)T + (-0.876 - 0.481i)T^{2} \)
97 \( 1 + (-0.967 - 1.42i)T + (-0.368 + 0.929i)T^{2} \)
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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

−8.973669473695522732098861402673, −8.014590065692429131376665909898, −7.00686224133258212957669395623, −6.53430214076050150951604453849, −6.30343617221504469566414254444, −4.67011631617741392323696221739, −4.47527066066928438649755690229, −3.27776483847249724581137080712, −2.37001674789486341208953412429, −1.52636424606564721469198502409, 0.76765634995372156350424394191, 2.03379530163509663049968379230, 2.75672258975497628142650564088, 4.20768919979889126216358642908, 4.63529501759760337179274691587, 5.44801117588648594838314027708, 6.17967775155053456865133838244, 7.07929874153631908741597665726, 7.79645269002367685529669030908, 8.553809278810817404463997991181

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