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Properties

Label 2-4000-25.2-c0-0-0
Degree $2$
Conductor $4000$
Sign $0.994 + 0.105i$
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  + (0.587 − 0.809i)9-s + (1.76 − 0.278i)13-s + (−0.896 + 1.76i)17-s + (1.11 + 0.363i)29-s + (0.0489 + 0.309i)37-s + (−1.53 − 1.11i)41-s − i·49-s + (0.412 + 0.809i)53-s + (1.53 − 1.11i)61-s + (0.142 − 0.896i)73-s + (−0.309 − 0.951i)81-s + (0.690 + 0.951i)89-s + (−0.278 + 0.142i)97-s + 1.61·101-s + (−0.363 + 0.5i)109-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)9-s + (1.76 − 0.278i)13-s + (−0.896 + 1.76i)17-s + (1.11 + 0.363i)29-s + (0.0489 + 0.309i)37-s + (−1.53 − 1.11i)41-s − i·49-s + (0.412 + 0.809i)53-s + (1.53 − 1.11i)61-s + (0.142 − 0.896i)73-s + (−0.309 − 0.951i)81-s + (0.690 + 0.951i)89-s + (−0.278 + 0.142i)97-s + 1.61·101-s + (−0.363 + 0.5i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.994 + 0.105i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (2657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.994 + 0.105i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.465384419\)
\(L(\frac12)\) \(\approx\) \(1.465384419\)
\(L(1)\) 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 \( 1 + (-0.587 + 0.809i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (-1.76 + 0.278i)T + (0.951 - 0.309i)T^{2} \)
17 \( 1 + (0.896 - 1.76i)T + (-0.587 - 0.809i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.951 - 0.309i)T^{2} \)
29 \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.0489 - 0.309i)T + (-0.951 + 0.309i)T^{2} \)
41 \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.587 - 0.809i)T^{2} \)
53 \( 1 + (-0.412 - 0.809i)T + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (-0.587 - 0.809i)T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.142 + 0.896i)T + (-0.951 - 0.309i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.587 + 0.809i)T^{2} \)
89 \( 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.278 - 0.142i)T + (0.587 - 0.809i)T^{2} \)
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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

−8.566934501482305945741206327626, −8.140434587313513454323523124661, −6.91291594940670005380097965151, −6.45012110743747835061881979040, −5.85173660139586633968019129454, −4.78554773091469377040526984307, −3.78078493873032308245740174740, −3.52244590849793936171477582073, −2.02921698656194227111275542751, −1.09943522073885087795436848309, 1.11519974014613452199786838663, 2.21647296787341595526501109389, 3.15585883393548520857470050963, 4.20064040665824189601359278718, 4.79155203502633540688866692971, 5.64562952951277556262959605093, 6.61958689752670733596629044534, 7.02519498708550208598214563735, 8.024996762893183696577410326444, 8.572826510963427494698232511226

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