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Properties

Label 2-4000-40.19-c0-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(1.737685819\)
\(L(\frac12)\) \(\approx\) \(1.737685819\)
\(L(1)\) 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

−8.470259163068633184176072872077, −8.104603832757456008671334297611, −7.05210573781778434121091423480, −6.40197976500528094166704712810, −5.88413258055005476587024174285, −4.62794526465199548114984777823, −4.06911110695905757244239574780, −3.45457170588125784975442313407, −1.82373677731878841011899288934, −1.35371354344164158694770612844, 1.35371354344164158694770612844, 1.82373677731878841011899288934, 3.45457170588125784975442313407, 4.06911110695905757244239574780, 4.62794526465199548114984777823, 5.88413258055005476587024174285, 6.40197976500528094166704712810, 7.05210573781778434121091423480, 8.104603832757456008671334297611, 8.470259163068633184176072872077

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