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Properties

Label 2-4000-20.19-c0-0-2
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  + 0.618·3-s + 7-s − 0.618·9-s + i·11-s + 0.618i·13-s + i·17-s − 0.618i·19-s + 0.618·21-s − 27-s + 29-s + 1.61i·31-s + 0.618i·33-s + 0.381i·39-s + 41-s − 43-s + ⋯
L(s)  = 1  + 0.618·3-s + 7-s − 0.618·9-s + i·11-s + 0.618i·13-s + i·17-s − 0.618i·19-s + 0.618·21-s − 27-s + 29-s + 1.61i·31-s + 0.618i·33-s + 0.381i·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

\(L(\frac{1}{2})\) \(\approx\) \(1.665965829\)
\(L(\frac12)\) \(\approx\) \(1.665965829\)
\(L(1)\) 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 \( 1 - 0.618T + T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 - iT - T^{2} \)
13 \( 1 - 0.618iT - T^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 + 0.618iT - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 - 1.61iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 - 1.61T + T^{2} \)
53 \( 1 + 1.61iT - T^{2} \)
59 \( 1 - 1.61iT - T^{2} \)
61 \( 1 - 0.618T + T^{2} \)
67 \( 1 + 1.61T + T^{2} \)
71 \( 1 - iT - T^{2} \)
73 \( 1 + 1.61iT - T^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 1.61iT - T^{2} \)
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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

−8.740188626618862092852198141463, −8.090316399249803900549980951911, −7.35896025372094302524491186733, −6.65171439166113943830585481667, −5.70196599494598929383165716967, −4.81249691252896212866574397069, −4.26551612399996911637692952806, −3.20684500268021427538012663884, −2.26584819545634445180300634459, −1.51214057368935250304377143759, 0.909903505721793999753753277241, 2.28672155570065675977511894169, 2.95431098526613991762991281851, 3.84446131860016018738678270684, 4.78882258062993015213640737377, 5.62084944833434498516951877002, 6.14249463876937981513856908204, 7.37289842571291583557035392842, 7.965084578149038080305903315515, 8.405912234660421703102026341689

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