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Properties

Label 2-4000-100.71-c0-0-1
Degree $2$
Conductor $4000$
Sign $-0.218 + 0.975i$
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)3-s − 0.618i·7-s + (−0.190 − 0.587i)13-s + (−0.951 − 1.30i)19-s + (0.500 + 0.363i)21-s + (−0.951 − 0.309i)23-s + (−0.951 − 0.309i)27-s + (−1.30 − 0.951i)29-s + (−0.587 − 0.809i)31-s + (0.309 + 0.951i)37-s + (0.587 + 0.190i)39-s + 1.61i·43-s + (0.363 − 0.5i)47-s + 0.618·49-s + (−0.809 − 0.587i)53-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)3-s − 0.618i·7-s + (−0.190 − 0.587i)13-s + (−0.951 − 1.30i)19-s + (0.500 + 0.363i)21-s + (−0.951 − 0.309i)23-s + (−0.951 − 0.309i)27-s + (−1.30 − 0.951i)29-s + (−0.587 − 0.809i)31-s + (0.309 + 0.951i)37-s + (0.587 + 0.190i)39-s + 1.61i·43-s + (0.363 − 0.5i)47-s + 0.618·49-s + (−0.809 − 0.587i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.218 + 0.975i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ -0.218 + 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4221169199\)
\(L(\frac12)\) \(\approx\) \(0.4221169199\)
\(L(1)\) 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 \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \)
7 \( 1 + 0.618iT - T^{2} \)
11 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \)
29 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T^{2} \)
43 \( 1 - 1.61iT - T^{2} \)
47 \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.309 + 0.951i)T^{2} \)
71 \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
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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

−8.313499975448239899849433853385, −7.74935139754921637088016768507, −6.92790392731338462199450633641, −6.04986637202879524979403857795, −5.43245289164208157077884388030, −4.38175614992658115751493001002, −4.24646235589071134660338899228, −2.99843782986114989786713229061, −1.93401610466446776177180549580, −0.24087963768786083539431845952, 1.55341484233124593480982043600, 2.13623862045543023554874664814, 3.51656931982201179403217121866, 4.22738752948814305117997423685, 5.48253445940003843802495563250, 5.84647250114383527713010531185, 6.60219322402632972673554619464, 7.33237182223157340463789738687, 7.919817914668191437967980547287, 8.963973019650543863098148949657

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