↑

Properties

Label 2-4000-100.11-c0-0-0
Degree $2$
Conductor $4000$
Sign $-0.999 - 0.0314i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.951 − 0.309i)3-s + 1.61i·7-s + (−1.30 − 0.951i)13-s + (0.587 − 0.190i)19-s + (0.500 − 1.53i)21-s + (0.587 + 0.809i)23-s + (0.587 + 0.809i)27-s + (−0.190 + 0.587i)29-s + (−0.951 + 0.309i)31-s + (−0.809 − 0.587i)37-s + (0.951 + 1.30i)39-s − 0.618i·43-s + (−1.53 − 0.5i)47-s − 1.61·49-s + (0.309 − 0.951i)53-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)3-s + 1.61i·7-s + (−1.30 − 0.951i)13-s + (0.587 − 0.190i)19-s + (0.500 − 1.53i)21-s + (0.587 + 0.809i)23-s + (0.587 + 0.809i)27-s + (−0.190 + 0.587i)29-s + (−0.951 + 0.309i)31-s + (−0.809 − 0.587i)37-s + (0.951 + 1.30i)39-s − 0.618i·43-s + (−1.53 − 0.5i)47-s − 1.61·49-s + (0.309 − 0.951i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.999 - 0.0314i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (3551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ -0.999 - 0.0314i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07958721884\)
\(L(\frac12)\) \(\approx\) \(0.07958721884\)
\(L(1)\) 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 \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \)
7 \( 1 - 1.61iT - T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
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 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (0.309 + 0.951i)T^{2} \)
43 \( 1 + 0.618iT - T^{2} \)
47 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \)
89 \( 1 + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.074023086557614699147540904779, −8.277458192685881202772343885862, −7.36420463046054766811379354058, −6.78877605204185543499508796100, −5.78277707737691653566252902609, −5.33292846633352294815599236821, −5.01726529161328919171130668490, −3.39053697085566050727959757582, −2.71272672124375574960326651375, −1.62156842507064841215760385504, 0.05019045340467301331952415142, 1.43613623158019832408869673402, 2.76468884492794320996975149634, 3.88285999558255920502384949977, 4.64213226703525501538592643446, 5.03380355354203972612798253189, 6.11489038109525781743010099301, 6.82487252579703178784931173179, 7.38678354908023874844592639001, 8.072711788058518959231121540439

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