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Properties

Label 2-4000-100.39-c0-0-3
Degree $2$
Conductor $4000$
Sign $0.498 + 0.866i$
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  + (0.309 − 0.951i)3-s + 1.61·7-s + (0.951 − 1.30i)13-s + (−0.587 + 0.190i)19-s + (0.500 − 1.53i)21-s + (−0.809 + 0.587i)23-s + (0.809 − 0.587i)27-s + (0.190 − 0.587i)29-s + (−0.951 + 0.309i)31-s + (−0.587 + 0.809i)37-s + (−0.951 − 1.30i)39-s + 0.618·43-s + (−0.5 + 1.53i)47-s + 1.61·49-s + (0.951 + 0.309i)53-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)3-s + 1.61·7-s + (0.951 − 1.30i)13-s + (−0.587 + 0.190i)19-s + (0.500 − 1.53i)21-s + (−0.809 + 0.587i)23-s + (0.809 − 0.587i)27-s + (0.190 − 0.587i)29-s + (−0.951 + 0.309i)31-s + (−0.587 + 0.809i)37-s + (−0.951 − 1.30i)39-s + 0.618·43-s + (−0.5 + 1.53i)47-s + 1.61·49-s + (0.951 + 0.309i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.498 + 0.866i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (3199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.498 + 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.852113061\)
\(L(\frac12)\) \(\approx\) \(1.852113061\)
\(L(1)\) 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 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 - 1.61T + T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
29 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.309 + 0.951i)T^{2} \)
43 \( 1 - 0.618T + T^{2} \)
47 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (-0.809 + 0.587i)T^{2} \)
71 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \)
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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

−8.337278805969994546519825837253, −7.76266676546745900336736965414, −7.38465477145024823501429647181, −6.26041191177197082492048374968, −5.63872156075381401222679002498, −4.76593201985756159835433121293, −3.94000938596300812377468758890, −2.81318755090589922217851250374, −1.80453810769698457888590891357, −1.21359031867850307218632711946, 1.49102229140475221554055949740, 2.24978459082322600216950942499, 3.70951708029968022373038195566, 4.15129321246153231689223751688, 4.80495157789258978762816413539, 5.59368679924210109415420085198, 6.59757824707665936752858036506, 7.33452253350922812040502561672, 8.283815195636341383433668628851, 8.808931071875371729101214178564

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