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Properties

Label 2-4000-125.3-c0-0-0
Degree $2$
Conductor $4000$
Sign $0.963 - 0.266i$
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  + (0.684 + 0.728i)5-s + (0.998 − 0.0627i)9-s + (0.743 − 0.843i)13-s + (1.41 + 0.508i)17-s + (−0.0627 + 0.998i)25-s + (−0.742 − 1.35i)29-s + (−1.01 − 0.0958i)37-s + (−1.75 − 0.450i)41-s + (0.728 + 0.684i)45-s + (−0.951 + 0.309i)49-s + (0.313 − 0.461i)53-s + (1.32 − 0.340i)61-s + (1.12 − 0.0353i)65-s + (0.627 + 1.45i)73-s + (0.992 − 0.125i)81-s + ⋯
L(s)  = 1  + (0.684 + 0.728i)5-s + (0.998 − 0.0627i)9-s + (0.743 − 0.843i)13-s + (1.41 + 0.508i)17-s + (−0.0627 + 0.998i)25-s + (−0.742 − 1.35i)29-s + (−1.01 − 0.0958i)37-s + (−1.75 − 0.450i)41-s + (0.728 + 0.684i)45-s + (−0.951 + 0.309i)49-s + (0.313 − 0.461i)53-s + (1.32 − 0.340i)61-s + (1.12 − 0.0353i)65-s + (0.627 + 1.45i)73-s + (0.992 − 0.125i)81-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.963 - 0.266i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (2753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.963 - 0.266i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.714943846\)
\(L(\frac12)\) \(\approx\) \(1.714943846\)
\(L(1)\) 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 \( 1 + (-0.684 - 0.728i)T \)
good3 \( 1 + (-0.998 + 0.0627i)T^{2} \)
7 \( 1 + (0.951 - 0.309i)T^{2} \)
11 \( 1 + (-0.425 + 0.904i)T^{2} \)
13 \( 1 + (-0.743 + 0.843i)T + (-0.125 - 0.992i)T^{2} \)
17 \( 1 + (-1.41 - 0.508i)T + (0.770 + 0.637i)T^{2} \)
19 \( 1 + (-0.0627 + 0.998i)T^{2} \)
23 \( 1 + (0.481 + 0.876i)T^{2} \)
29 \( 1 + (0.742 + 1.35i)T + (-0.535 + 0.844i)T^{2} \)
31 \( 1 + (-0.637 + 0.770i)T^{2} \)
37 \( 1 + (1.01 + 0.0958i)T + (0.982 + 0.187i)T^{2} \)
41 \( 1 + (1.75 + 0.450i)T + (0.876 + 0.481i)T^{2} \)
43 \( 1 + (-0.587 - 0.809i)T^{2} \)
47 \( 1 + (0.248 - 0.968i)T^{2} \)
53 \( 1 + (-0.313 + 0.461i)T + (-0.368 - 0.929i)T^{2} \)
59 \( 1 + (-0.728 + 0.684i)T^{2} \)
61 \( 1 + (-1.32 + 0.340i)T + (0.876 - 0.481i)T^{2} \)
67 \( 1 + (0.844 - 0.535i)T^{2} \)
71 \( 1 + (0.968 + 0.248i)T^{2} \)
73 \( 1 + (-0.627 - 1.45i)T + (-0.684 + 0.728i)T^{2} \)
79 \( 1 + (-0.0627 - 0.998i)T^{2} \)
83 \( 1 + (0.998 + 0.0627i)T^{2} \)
89 \( 1 + (-0.621 + 1.57i)T + (-0.728 - 0.684i)T^{2} \)
97 \( 1 + (0.512 - 1.76i)T + (-0.844 - 0.535i)T^{2} \)
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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

−8.553860130376247507126641745329, −7.889443828499257562153304375671, −7.15881450732609756463210165828, −6.45752506459838043753464141475, −5.72079247636749501219321714678, −5.11752931329105590825945227523, −3.79299318466904046147138019612, −3.38373275366176879959244188837, −2.17023577444957140787254925945, −1.27043454777770090795485033963, 1.28238903988601333664174999939, 1.82116895637473185427874998180, 3.24442687701479249421595802138, 4.03098322699275004701524266712, 5.02130723905017296570899778406, 5.42221403734480733192675525610, 6.49751364057975111085009532544, 7.01583031967968420876403850497, 7.935076902151283594340775454191, 8.674375553753824008011697437692

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