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Properties

Label 2-4000-20.19-c0-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(0.7136560652\)
\(L(\frac12)\) \(\approx\) \(0.7136560652\)
\(L(1)\) 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

−8.394327977431277062302897247324, −8.016914025121238575921135713001, −6.75438530871406201793612599014, −6.20463864995257930710152055263, −5.85898271661021895350604121490, −4.86860479767716021959312624812, −3.82249651758657398471140881310, −3.20110576670398805653953708615, −2.10114680725156623572355602719, −0.56015373163487400013047920732, 0.925627268605791864753136936526, 2.61115224021243270133856994796, 3.05823477311892263798481465006, 4.31100323076486868579849804436, 5.07945304050277701269553633080, 5.71249726665671206081542296782, 6.64513256481609007237764896828, 6.99126317476544367256361640332, 7.936183120022798599922605744164, 8.834737769125314783536093371264

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