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Properties

Label 2-4000-100.19-c0-0-3
Degree $2$
Conductor $4000$
Sign $0.698 + 0.715i$
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  + (0.809 − 0.587i)3-s + 0.618·7-s + (−0.587 − 0.190i)13-s + (0.951 − 1.30i)19-s + (0.500 − 0.363i)21-s + (−0.309 − 0.951i)23-s + (0.309 + 0.951i)27-s + (1.30 − 0.951i)29-s + (−0.587 + 0.809i)31-s + (−0.951 − 0.309i)37-s + (−0.587 + 0.190i)39-s + 1.61·43-s + (0.5 − 0.363i)47-s − 0.618·49-s + (−0.587 − 0.809i)53-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)3-s + 0.618·7-s + (−0.587 − 0.190i)13-s + (0.951 − 1.30i)19-s + (0.500 − 0.363i)21-s + (−0.309 − 0.951i)23-s + (0.309 + 0.951i)27-s + (1.30 − 0.951i)29-s + (−0.587 + 0.809i)31-s + (−0.951 − 0.309i)37-s + (−0.587 + 0.190i)39-s + 1.61·43-s + (0.5 − 0.363i)47-s − 0.618·49-s + (−0.587 − 0.809i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.698 + 0.715i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.698 + 0.715i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.827189647\)
\(L(\frac12)\) \(\approx\) \(1.827189647\)
\(L(1)\) 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 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 - 0.618T + T^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - 1.61T + T^{2} \)
47 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \)
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 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \)
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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

−8.459901364321185161883288281042, −7.85175947329805130677559817954, −7.18411010514597861425386976152, −6.61812873056241131956002142237, −5.37005959172990458988840389025, −4.89045013640024652080850649079, −3.86812732928783007107415056218, −2.70033166724542125463324187060, −2.32962611307530444105927543060, −1.05172027443866975910490890677, 1.38838509443711431897979746616, 2.45972849957572861003855248015, 3.39915574596050139354424337928, 3.99001028158152285685421755650, 4.92108301208316290864223257998, 5.60642798076725497610531163126, 6.55181341291492427110723619922, 7.54701949969936751700860934806, 7.992504440880373676808241487813, 8.746013749515391522983209200314

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