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Properties

Label 2-4000-5.2-c0-0-5
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

\(L(\frac{1}{2})\) \(\approx\) \(2.329854572\)
\(L(\frac12)\) \(\approx\) \(2.329854572\)
\(L(1)\) 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

−8.375186422886460125953510151014, −8.036398130104487138981940247451, −7.24966132290250130407037079757, −6.52023816975397895382246619042, −5.60318488986427033278843789179, −4.89236883880019354782991173469, −3.66226756114262892537226229285, −2.75725850041609844429806866561, −2.01962816610871649754267042469, −1.45354841519763370959910470016, 1.45675138784089461243664648239, 2.47311108177770939946814574715, 3.51836237539743839272125161706, 4.14465410077724233788251791159, 4.67162333262897785774625084701, 5.35227071754596818969990478231, 6.73853596102398995778141886672, 7.61347804216057074765949989629, 8.110408232690932929831577313746, 8.582411398022748916432455911598

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