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Properties

Label 2-4000-40.19-c0-0-1
Degree $2$
Conductor $4000$
Sign $1$
Analytic cond. $1.99626$
Root an. cond. $1.41289$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.618·7-s + 9-s + 1.61·11-s − 1.61·13-s − 0.618·19-s + 1.61·23-s + 0.618·37-s + 0.618·41-s + 1.61·47-s − 0.618·49-s + 0.618·53-s − 0.618·59-s − 0.618·63-s − 1.00·77-s + 81-s − 1.61·89-s + 1.00·91-s + 1.61·99-s + 1.61·103-s − 1.61·117-s + ⋯
L(s)  = 1  − 0.618·7-s + 9-s + 1.61·11-s − 1.61·13-s − 0.618·19-s + 1.61·23-s + 0.618·37-s + 0.618·41-s + 1.61·47-s − 0.618·49-s + 0.618·53-s − 0.618·59-s − 0.618·63-s − 1.00·77-s + 81-s − 1.61·89-s + 1.00·91-s + 1.61·99-s + 1.61·103-s − 1.61·117-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (1999, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.370148626\)
\(L(\frac12)\) \(\approx\) \(1.370148626\)
\(L(1)\) 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 \( 1 - T^{2} \)
7 \( 1 + 0.618T + T^{2} \)
11 \( 1 - 1.61T + T^{2} \)
13 \( 1 + 1.61T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 0.618T + T^{2} \)
23 \( 1 - 1.61T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 0.618T + T^{2} \)
41 \( 1 - 0.618T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 1.61T + T^{2} \)
53 \( 1 - 0.618T + T^{2} \)
59 \( 1 + 0.618T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 1.61T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.881577073493336601636099886802, −7.70719294133735193083891291819, −6.99766707196473639624848527592, −6.69611018819678621099816586956, −5.72535084843712405811691345050, −4.65348080164546101045168717668, −4.19208158143370062057133238775, −3.19483955427911074583385104823, −2.20705075468631241711432618609, −1.04328909854902685539345275255, 1.04328909854902685539345275255, 2.20705075468631241711432618609, 3.19483955427911074583385104823, 4.19208158143370062057133238775, 4.65348080164546101045168717668, 5.72535084843712405811691345050, 6.69611018819678621099816586956, 6.99766707196473639624848527592, 7.70719294133735193083891291819, 8.881577073493336601636099886802

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