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Properties

Label 2-4000-4.3-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  + 1.61i·3-s + i·7-s − 1.61·9-s − i·11-s + 1.61·13-s + 17-s + 1.61i·19-s − 1.61·21-s − i·27-s − 29-s + 0.618i·31-s + 1.61·33-s + 2.61i·39-s + 41-s + i·43-s + ⋯
L(s)  = 1  + 1.61i·3-s + i·7-s − 1.61·9-s − i·11-s + 1.61·13-s + 17-s + 1.61i·19-s − 1.61·21-s − i·27-s − 29-s + 0.618i·31-s + 1.61·33-s + 2.61i·39-s + 41-s + i·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: $\chi_{4000} (2751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.350073088\)
\(L(\frac12)\) \(\approx\) \(1.350073088\)
\(L(1)\) 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 \( 1 - 1.61iT - T^{2} \)
7 \( 1 - iT - T^{2} \)
11 \( 1 + iT - T^{2} \)
13 \( 1 - 1.61T + T^{2} \)
17 \( 1 - T + T^{2} \)
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 \( 1 - iT - T^{2} \)
47 \( 1 + 0.618iT - T^{2} \)
53 \( 1 + 0.618T + T^{2} \)
59 \( 1 + 0.618iT - T^{2} \)
61 \( 1 + 1.61T + T^{2} \)
67 \( 1 - 0.618iT - T^{2} \)
71 \( 1 + iT - T^{2} \)
73 \( 1 + 0.618T + T^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - 0.618T + T^{2} \)
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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

−9.049031713650509966180474118448, −8.355578984449951290210364343077, −7.81195743629333537415758356538, −6.19498726305726455646597250326, −5.83183253709136063028482332506, −5.30534968837143397432003601201, −4.21910054316404716766348125082, −3.48322579645655121941470518930, −3.09020815438981819715662561236, −1.53465440101401524412431320365, 0.839094865532504771603129335201, 1.58655329027364066929084921440, 2.61148002864408513649596611318, 3.65894834259825561455004339511, 4.52302139786196214838776896815, 5.67213465145316684833732265330, 6.29911330210018601226373462263, 7.13903009919667631776509442086, 7.39737675226725226777886458798, 8.087781238136283289135621862359

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