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Properties

Label 2-4000-5.2-c0-0-0
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  + (−1.26 + 1.26i)3-s + (−1.39 − 1.39i)7-s − 2.17i·9-s + 3.52·21-s + (0.221 − 0.221i)23-s + (1.48 + 1.48i)27-s + 0.618i·29-s − 1.90·41-s + (−0.642 + 0.642i)43-s + (0.642 + 0.642i)47-s + 2.90i·49-s + 1.17·61-s + (−3.03 + 3.03i)63-s + (1 + i)67-s + 0.557i·69-s + ⋯
L(s)  = 1  + (−1.26 + 1.26i)3-s + (−1.39 − 1.39i)7-s − 2.17i·9-s + 3.52·21-s + (0.221 − 0.221i)23-s + (1.48 + 1.48i)27-s + 0.618i·29-s − 1.90·41-s + (−0.642 + 0.642i)43-s + (0.642 + 0.642i)47-s + 2.90i·49-s + 1.17·61-s + (−3.03 + 3.03i)63-s + (1 + i)67-s + 0.557i·69-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: $\chi_{4000} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3199845255\)
\(L(\frac12)\) \(\approx\) \(0.3199845255\)
\(L(1)\) 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 \( 1 + (1.26 - 1.26i)T - iT^{2} \)
7 \( 1 + (1.39 + 1.39i)T + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.221 + 0.221i)T - iT^{2} \)
29 \( 1 - 0.618iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + 1.90T + T^{2} \)
43 \( 1 + (0.642 - 0.642i)T - iT^{2} \)
47 \( 1 + (-0.642 - 0.642i)T + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.17T + T^{2} \)
67 \( 1 + (-1 - i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (1.39 - 1.39i)T - iT^{2} \)
89 \( 1 - 1.61iT - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.270304715018387424574097355865, −8.255695267632587856124295435541, −6.88789690545511710472893475375, −6.83057169047871306989968210588, −5.88271736176462158656774225989, −5.14141223076914320897709998904, −4.30894713942810936620461254597, −3.74805046473614998876490977834, −3.03770095048705196143419061415, −0.988672845703823806491531230694, 0.25827632721229580173528320062, 1.76843414174637629651620945729, 2.57166316460757302328490305826, 3.58228996237485041420771946904, 5.06784506880250345709478100931, 5.53435653045961455455190985503, 6.26048022306036906947191566709, 6.67002544999434137894540730662, 7.34486413387069622796353076511, 8.327830300033225574652588455627

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