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Properties

Label 2-10e3-200.189-c1-0-20
Degree $2$
Conductor $1000$
Sign $-0.903 - 0.429i$
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  + (−0.345 + 1.37i)2-s + (−0.615 + 1.89i)3-s + (−1.76 − 0.947i)4-s + (−2.38 − 1.49i)6-s − 1.38i·7-s + (1.90 − 2.08i)8-s + (−0.783 − 0.568i)9-s + (3.04 + 4.18i)11-s + (2.87 − 2.75i)12-s + (2.49 + 1.81i)13-s + (1.89 + 0.477i)14-s + (2.20 + 3.33i)16-s + (3.63 − 1.17i)17-s + (1.05 − 0.877i)18-s + (6.09 − 1.97i)19-s + ⋯
L(s)  = 1  + (−0.244 + 0.969i)2-s + (−0.355 + 1.09i)3-s + (−0.880 − 0.473i)4-s + (−0.973 − 0.611i)6-s − 0.521i·7-s + (0.674 − 0.738i)8-s + (−0.261 − 0.189i)9-s + (0.917 + 1.26i)11-s + (0.831 − 0.794i)12-s + (0.692 + 0.503i)13-s + (0.506 + 0.127i)14-s + (0.550 + 0.834i)16-s + (0.880 − 0.286i)17-s + (0.247 − 0.206i)18-s + (1.39 − 0.454i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 - 0.429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.903 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1000\)    =    \(2^{3} \cdot 5^{3}\)
Sign: $-0.903 - 0.429i$
Analytic conductor: \(7.98504\)
Root analytic conductor: \(2.82578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1000} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1000,\ (\ :1/2),\ -0.903 - 0.429i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.277381 + 1.22921i\)
\(L(\frac12)\) \(\approx\) \(0.277381 + 1.22921i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.345 - 1.37i)T \)
5 \( 1 \)
good3 \( 1 + (0.615 - 1.89i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + 1.38iT - 7T^{2} \)
11 \( 1 + (-3.04 - 4.18i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-2.49 - 1.81i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.63 + 1.17i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-6.09 + 1.97i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.168 - 0.232i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (8.75 + 2.84i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.38 - 4.27i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.07 + 2.96i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-3.28 - 2.38i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 7.54T + 43T^{2} \)
47 \( 1 + (0.954 + 0.310i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.02 - 3.15i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (1.05 - 1.44i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.30 + 1.80i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (0.552 + 1.69i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (4.68 - 14.4i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (6.88 + 9.46i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.381 - 1.17i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.29 - 3.99i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (9.13 - 6.63i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-4.65 - 1.51i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.00461338735079763828626022465, −9.499305211712014398918448429601, −8.951540151627910750249902808127, −7.47309028121737834082085780253, −7.20732926608925592491071783091, −5.97248149908090305481879680817, −5.14463623448748986378631257193, −4.30214737182291330490651300834, −3.67914018973172445577213770109, −1.33204903721725801531738267821, 0.824513162344981120180365861134, 1.63934772439784243594338316802, 3.12261488071577288943612317120, 3.85304914104525818504711878915, 5.58421261279949772390701885318, 5.94218899405706741435561012152, 7.31839053012607099866601865629, 8.006048539770744626012859711607, 8.880889609469664258410136991633, 9.573465044980592037925136495381

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