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Properties

Label 2-4000-20.19-c0-0-4
Degree $2$
Conductor $4000$
Sign $0.707 - 0.707i$
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  + 1.61·3-s − 7-s + 1.61·9-s + i·11-s + 1.61i·13-s − i·17-s + 1.61i·19-s − 1.61·21-s + 27-s + 29-s − 0.618i·31-s + 1.61i·33-s + 2.61i·39-s + 41-s + 43-s + ⋯
L(s)  = 1  + 1.61·3-s − 7-s + 1.61·9-s + i·11-s + 1.61i·13-s − i·17-s + 1.61i·19-s − 1.61·21-s + 27-s + 29-s − 0.618i·31-s + 1.61i·33-s + 2.61i·39-s + 41-s + 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (3999, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.048278016\)
\(L(\frac12)\) \(\approx\) \(2.048278016\)
\(L(1)\) 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 \( 1 - 1.61T + T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 - iT - T^{2} \)
13 \( 1 - 1.61iT - T^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 - 1.61iT - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + 0.618iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 - 0.618T + T^{2} \)
53 \( 1 + 0.618iT - T^{2} \)
59 \( 1 + 0.618iT - T^{2} \)
61 \( 1 + 1.61T + T^{2} \)
67 \( 1 + 0.618T + T^{2} \)
71 \( 1 - iT - T^{2} \)
73 \( 1 + 0.618iT - T^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 0.618iT - T^{2} \)
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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

−8.895748905423115336924933361371, −7.951982087989580336423013052568, −7.38043938906018656482689515789, −6.74826300414399148668911039043, −5.94857173885331190257579793339, −4.55271782624030990142232874972, −4.06913072878121017127986767255, −3.19869062366966403065515422151, −2.41751880719946100298273563042, −1.65066807237253700638026933202, 0.971971759369874899390028672324, 2.57841208149631244140633655939, 2.96969872843744855749282846253, 3.58649154696018631387242898961, 4.53268364660308515615670778512, 5.70502236503410971329086331721, 6.37222412096649869793343940357, 7.31627290886217409658214771400, 7.930351716594804099982321756886, 8.632681199601815985390875597518

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