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Properties

Label 2-4000-125.27-c0-0-0
Degree $2$
Conductor $4000$
Sign $0.812 - 0.582i$
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  + (−0.770 + 0.637i)5-s + (0.982 − 0.187i)9-s + (1.61 + 1.09i)13-s + (−0.572 − 0.967i)17-s + (0.187 − 0.982i)25-s + (−0.961 − 0.0604i)29-s + (1.91 + 0.557i)37-s + (−0.362 − 0.340i)41-s + (−0.637 + 0.770i)45-s + (−0.587 + 0.809i)49-s + (1.46 + 0.327i)53-s + (−1.12 + 1.05i)61-s + (−1.94 + 0.183i)65-s + (0.747 + 0.269i)73-s + (0.929 − 0.368i)81-s + ⋯
L(s)  = 1  + (−0.770 + 0.637i)5-s + (0.982 − 0.187i)9-s + (1.61 + 1.09i)13-s + (−0.572 − 0.967i)17-s + (0.187 − 0.982i)25-s + (−0.961 − 0.0604i)29-s + (1.91 + 0.557i)37-s + (−0.362 − 0.340i)41-s + (−0.637 + 0.770i)45-s + (−0.587 + 0.809i)49-s + (1.46 + 0.327i)53-s + (−1.12 + 1.05i)61-s + (−1.94 + 0.183i)65-s + (0.747 + 0.269i)73-s + (0.929 − 0.368i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.812 - 0.582i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (3777, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.812 - 0.582i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.282970459\)
\(L(\frac12)\) \(\approx\) \(1.282970459\)
\(L(1)\) 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 \( 1 + (0.770 - 0.637i)T \)
good3 \( 1 + (-0.982 + 0.187i)T^{2} \)
7 \( 1 + (0.587 - 0.809i)T^{2} \)
11 \( 1 + (0.968 - 0.248i)T^{2} \)
13 \( 1 + (-1.61 - 1.09i)T + (0.368 + 0.929i)T^{2} \)
17 \( 1 + (0.572 + 0.967i)T + (-0.481 + 0.876i)T^{2} \)
19 \( 1 + (0.187 - 0.982i)T^{2} \)
23 \( 1 + (-0.998 - 0.0627i)T^{2} \)
29 \( 1 + (0.961 + 0.0604i)T + (0.992 + 0.125i)T^{2} \)
31 \( 1 + (0.876 + 0.481i)T^{2} \)
37 \( 1 + (-1.91 - 0.557i)T + (0.844 + 0.535i)T^{2} \)
41 \( 1 + (0.362 + 0.340i)T + (0.0627 + 0.998i)T^{2} \)
43 \( 1 + (0.951 - 0.309i)T^{2} \)
47 \( 1 + (-0.684 + 0.728i)T^{2} \)
53 \( 1 + (-1.46 - 0.327i)T + (0.904 + 0.425i)T^{2} \)
59 \( 1 + (0.637 + 0.770i)T^{2} \)
61 \( 1 + (1.12 - 1.05i)T + (0.0627 - 0.998i)T^{2} \)
67 \( 1 + (-0.125 - 0.992i)T^{2} \)
71 \( 1 + (0.728 + 0.684i)T^{2} \)
73 \( 1 + (-0.747 - 0.269i)T + (0.770 + 0.637i)T^{2} \)
79 \( 1 + (0.187 + 0.982i)T^{2} \)
83 \( 1 + (0.982 + 0.187i)T^{2} \)
89 \( 1 + (-0.226 + 0.106i)T + (0.637 - 0.770i)T^{2} \)
97 \( 1 + (-0.508 + 0.448i)T + (0.125 - 0.992i)T^{2} \)
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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

−8.779348864735560501882535914626, −7.81308056582872107806097616885, −7.23716567557638959794984083441, −6.57068792675382959158516740875, −5.99374602230841134776163813375, −4.64986773297721034487597131529, −4.11889651936653789605734057541, −3.43457272550018559574926831123, −2.33868706818668957962341424114, −1.15005556925190022366101250321, 0.915251977458336363664003567100, 1.90438809237090321312034974581, 3.39085183875951566862246323857, 3.93815851012958711866818439285, 4.65681516336287422401086859471, 5.61088716177402118493419900467, 6.29365993686702910527564906534, 7.22636326437236278535639905833, 8.003450029332120143021150074892, 8.356486904417852929057045946349

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