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Properties

Label 2-4000-125.8-c0-0-0
Degree $2$
Conductor $4000$
Sign $-0.782 + 0.622i$
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  + (0.125 − 0.992i)5-s + (−0.248 − 0.968i)9-s + (−1.72 − 1.01i)13-s + (0.842 + 0.572i)17-s + (−0.968 − 0.248i)25-s + (−0.666 + 0.313i)29-s + (0.400 − 0.173i)37-s + (−1.05 − 1.65i)41-s + (−0.992 + 0.125i)45-s + (0.951 + 0.309i)49-s + (−1.88 − 0.0591i)53-s + (0.134 − 0.211i)61-s + (−1.22 + 1.57i)65-s + (−1.27 + 1.44i)73-s + (−0.876 + 0.481i)81-s + ⋯
L(s)  = 1  + (0.125 − 0.992i)5-s + (−0.248 − 0.968i)9-s + (−1.72 − 1.01i)13-s + (0.842 + 0.572i)17-s + (−0.968 − 0.248i)25-s + (−0.666 + 0.313i)29-s + (0.400 − 0.173i)37-s + (−1.05 − 1.65i)41-s + (−0.992 + 0.125i)45-s + (0.951 + 0.309i)49-s + (−1.88 − 0.0591i)53-s + (0.134 − 0.211i)61-s + (−1.22 + 1.57i)65-s + (−1.27 + 1.44i)73-s + (−0.876 + 0.481i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.782 + 0.622i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1633, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ -0.782 + 0.622i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8247773599\)
\(L(\frac12)\) \(\approx\) \(0.8247773599\)
\(L(1)\) 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 \( 1 + (-0.125 + 0.992i)T \)
good3 \( 1 + (0.248 + 0.968i)T^{2} \)
7 \( 1 + (-0.951 - 0.309i)T^{2} \)
11 \( 1 + (-0.187 + 0.982i)T^{2} \)
13 \( 1 + (1.72 + 1.01i)T + (0.481 + 0.876i)T^{2} \)
17 \( 1 + (-0.842 - 0.572i)T + (0.368 + 0.929i)T^{2} \)
19 \( 1 + (-0.968 - 0.248i)T^{2} \)
23 \( 1 + (0.904 - 0.425i)T^{2} \)
29 \( 1 + (0.666 - 0.313i)T + (0.637 - 0.770i)T^{2} \)
31 \( 1 + (-0.929 + 0.368i)T^{2} \)
37 \( 1 + (-0.400 + 0.173i)T + (0.684 - 0.728i)T^{2} \)
41 \( 1 + (1.05 + 1.65i)T + (-0.425 + 0.904i)T^{2} \)
43 \( 1 + (0.587 - 0.809i)T^{2} \)
47 \( 1 + (0.844 - 0.535i)T^{2} \)
53 \( 1 + (1.88 + 0.0591i)T + (0.998 + 0.0627i)T^{2} \)
59 \( 1 + (0.992 + 0.125i)T^{2} \)
61 \( 1 + (-0.134 + 0.211i)T + (-0.425 - 0.904i)T^{2} \)
67 \( 1 + (-0.770 + 0.637i)T^{2} \)
71 \( 1 + (0.535 + 0.844i)T^{2} \)
73 \( 1 + (1.27 - 1.44i)T + (-0.125 - 0.992i)T^{2} \)
79 \( 1 + (-0.968 + 0.248i)T^{2} \)
83 \( 1 + (-0.248 + 0.968i)T^{2} \)
89 \( 1 + (-1.53 + 0.0967i)T + (0.992 - 0.125i)T^{2} \)
97 \( 1 + (1.41 + 0.508i)T + (0.770 + 0.637i)T^{2} \)
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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

−8.377192013658473576255507862583, −7.65433715031137168412077030050, −7.00123059806086901751890939469, −5.84283453230091013725837342642, −5.46861209338883607261939136422, −4.63947849703921643799146809367, −3.71805032495293898982217609571, −2.86142655314511113297705178004, −1.69226478609857905214421637167, −0.43121461283019072398784092311, 1.84339200967661911808172581391, 2.57002374035125197371261338539, 3.33123508310997123474964404715, 4.54848266261594598427541197890, 5.08889339954924696034205512266, 6.03411351791569229725673485142, 6.80787827771885772582820779392, 7.57398370570294520064771664106, 7.81949935701441323618394896775, 9.030484075928417017253857098557

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