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Properties

Label 2-10e3-200.21-c1-0-5
Degree $2$
Conductor $1000$
Sign $-0.999 + 0.00217i$
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  + (1.39 − 0.238i)2-s + (−1.01 + 1.39i)3-s + (1.88 − 0.665i)4-s + (−1.07 + 2.18i)6-s − 4.42·7-s + (2.47 − 1.37i)8-s + (0.0123 + 0.0380i)9-s + (−5.07 − 1.65i)11-s + (−0.981 + 3.29i)12-s + (0.730 − 0.237i)13-s + (−6.16 + 1.05i)14-s + (3.11 − 2.50i)16-s + (−4.60 + 3.34i)17-s + (0.0263 + 0.0501i)18-s + (−1.01 − 1.40i)19-s + ⋯
L(s)  = 1  + (0.985 − 0.168i)2-s + (−0.583 + 0.803i)3-s + (0.943 − 0.332i)4-s + (−0.439 + 0.890i)6-s − 1.67·7-s + (0.873 − 0.487i)8-s + (0.00412 + 0.0126i)9-s + (−1.53 − 0.497i)11-s + (−0.283 + 0.952i)12-s + (0.202 − 0.0658i)13-s + (−1.64 + 0.282i)14-s + (0.778 − 0.627i)16-s + (−1.11 + 0.811i)17-s + (0.00620 + 0.0118i)18-s + (−0.233 − 0.321i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00217i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00217i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1000\)    =    \(2^{3} \cdot 5^{3}\)
Sign: $-0.999 + 0.00217i$
Analytic conductor: \(7.98504\)
Root analytic conductor: \(2.82578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1000} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1000,\ (\ :1/2),\ -0.999 + 0.00217i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000399296 - 0.367728i\)
\(L(\frac12)\) \(\approx\) \(0.000399296 - 0.367728i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.238i)T \)
5 \( 1 \)
good3 \( 1 + (1.01 - 1.39i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + 4.42T + 7T^{2} \)
11 \( 1 + (5.07 + 1.65i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.730 + 0.237i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (4.60 - 3.34i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.01 + 1.40i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.62 - 5.01i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-1.21 + 1.66i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.10 - 1.53i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (6.18 - 2.01i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.754 + 2.32i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 6.65iT - 43T^{2} \)
47 \( 1 + (-6.50 - 4.72i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.74 - 3.77i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.02 + 0.334i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-6.86 - 2.23i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (3.72 + 5.12i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (-0.472 - 0.343i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.78 - 5.48i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (6.27 + 4.55i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.99 + 5.49i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (3.89 - 11.9i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (2.62 + 1.90i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.49628027909775656122778173244, −10.02159807589263240204615713560, −8.930970030228358210836423951632, −7.64876980003198776996893935027, −6.66595058905519495807233303220, −5.84141795557343549865570870428, −5.27907675488185399119797460327, −4.16954130660006562503575974876, −3.37882475087306019475353784681, −2.34700738674905288436469548032, 0.11462548921179558811277481473, 2.21001904323059920347897888603, 3.10265580712656651316647985701, 4.28740120340445380102191811043, 5.40168322380526677205747718473, 6.19214793863229985412778217757, 6.85448678807954736479942138035, 7.34782826371318404897874998983, 8.553466276360132984470356496211, 9.791990409841974162012542947806

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