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Properties

Label 2-10e3-200.3-c1-0-45
Degree $2$
Conductor $1000$
Sign $0.529 + 0.848i$
Analytic cond. $7.98504$
Root an. cond. $2.82578$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.221 + 1.39i)2-s + (−0.420 − 2.65i)3-s + (−1.90 + 0.618i)4-s + (3.61 − 1.17i)6-s + (0.557 + 0.557i)7-s + (−1.28 − 2.52i)8-s + (−4.02 + 1.30i)9-s + (−0.136 + 0.420i)11-s + (2.44 + 4.79i)12-s + (5.63 + 2.87i)13-s + (−0.655 + 0.902i)14-s + (3.23 − 2.35i)16-s + (0.857 − 5.41i)17-s + (−2.71 − 5.33i)18-s + (−2.07 − 2.85i)19-s + ⋯
L(s)  = 1  + (0.156 + 0.987i)2-s + (−0.242 − 1.53i)3-s + (−0.951 + 0.309i)4-s + (1.47 − 0.479i)6-s + (0.210 + 0.210i)7-s + (−0.453 − 0.891i)8-s + (−1.34 + 0.436i)9-s + (−0.0412 + 0.126i)11-s + (0.705 + 1.38i)12-s + (1.56 + 0.796i)13-s + (−0.175 + 0.241i)14-s + (0.809 − 0.587i)16-s + (0.208 − 1.31i)17-s + (−0.641 − 1.25i)18-s + (−0.476 − 0.655i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.529 + 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1000\)    =    \(2^{3} \cdot 5^{3}\)
Sign: $0.529 + 0.848i$
Analytic conductor: \(7.98504\)
Root analytic conductor: \(2.82578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1000} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1000,\ (\ :1/2),\ 0.529 + 0.848i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13597 - 0.630131i\)
\(L(\frac12)\) \(\approx\) \(1.13597 - 0.630131i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.221 - 1.39i)T \)
5 \( 1 \)
good3 \( 1 + (0.420 + 2.65i)T + (-2.85 + 0.927i)T^{2} \)
7 \( 1 + (-0.557 - 0.557i)T + 7iT^{2} \)
11 \( 1 + (0.136 - 0.420i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-5.63 - 2.87i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (-0.857 + 5.41i)T + (-16.1 - 5.25i)T^{2} \)
19 \( 1 + (2.07 + 2.85i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.568 - 1.11i)T + (-13.5 + 18.6i)T^{2} \)
29 \( 1 + (-2.33 - 1.69i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (5.90 + 8.13i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.33 + 4.58i)T + (-21.7 - 29.9i)T^{2} \)
41 \( 1 + (2.22 + 6.84i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (0.989 + 0.989i)T + 43iT^{2} \)
47 \( 1 + (-0.256 - 1.61i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (1.40 - 0.222i)T + (50.4 - 16.3i)T^{2} \)
59 \( 1 + (10.7 - 3.47i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.57 - 1.48i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (-0.986 + 6.22i)T + (-63.7 - 20.7i)T^{2} \)
71 \( 1 + (-6.33 + 8.71i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-10.8 + 5.50i)T + (42.9 - 59.0i)T^{2} \)
79 \( 1 + (10.5 + 7.68i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-13.6 - 2.15i)T + (78.9 + 25.6i)T^{2} \)
89 \( 1 + (-3.20 - 1.04i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (-0.281 + 0.0445i)T + (92.2 - 29.9i)T^{2} \)
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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

−9.341192705667252775527699397647, −8.815211884043447979782962902022, −7.86187090913423889854513565448, −7.24638883787420770319993644281, −6.51033134016321191367994617908, −5.88795716387599243236497889601, −4.89857756254433777983574474911, −3.63332884302758341633404086575, −2.08793574585524083654691493287, −0.65498608297886992983882041540, 1.39309214978211118921019797573, 3.19644522929416034777478138224, 3.77337340773017004770216446116, 4.57735327129258168183871818452, 5.52417095375179794393610634787, 6.23347965990230186758863461339, 8.269023218805029353728256082420, 8.530568063154600739772313247731, 9.649960868065911582881837512156, 10.29185699916744187821547881326

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