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Properties

Label 2-10e3-8.5-c1-0-60
Degree $2$
Conductor $1000$
Sign $0.231 + 0.972i$
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  + (−0.800 − 1.16i)2-s + 2.62i·3-s + (−0.719 + 1.86i)4-s + (3.06 − 2.10i)6-s − 0.269·7-s + (2.75 − 0.654i)8-s − 3.89·9-s − 4.58i·11-s + (−4.90 − 1.88i)12-s − 1.55i·13-s + (0.215 + 0.313i)14-s + (−2.96 − 2.68i)16-s − 0.609·17-s + (3.12 + 4.54i)18-s − 6.69i·19-s + ⋯
L(s)  = 1  + (−0.565 − 0.824i)2-s + 1.51i·3-s + (−0.359 + 0.933i)4-s + (1.25 − 0.858i)6-s − 0.101·7-s + (0.972 − 0.231i)8-s − 1.29·9-s − 1.38i·11-s + (−1.41 − 0.545i)12-s − 0.430i·13-s + (0.0575 + 0.0839i)14-s + (−0.741 − 0.671i)16-s − 0.147·17-s + (0.735 + 1.07i)18-s − 1.53i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1000\)    =    \(2^{3} \cdot 5^{3}\)
Sign: $0.231 + 0.972i$
Analytic conductor: \(7.98504\)
Root analytic conductor: \(2.82578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1000} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1000,\ (\ :1/2),\ 0.231 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.597556 - 0.472123i\)
\(L(\frac12)\) \(\approx\) \(0.597556 - 0.472123i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.800 + 1.16i)T \)
5 \( 1 \)
good3 \( 1 - 2.62iT - 3T^{2} \)
7 \( 1 + 0.269T + 7T^{2} \)
11 \( 1 + 4.58iT - 11T^{2} \)
13 \( 1 + 1.55iT - 13T^{2} \)
17 \( 1 + 0.609T + 17T^{2} \)
19 \( 1 + 6.69iT - 19T^{2} \)
23 \( 1 + 3.52T + 23T^{2} \)
29 \( 1 + 7.73iT - 29T^{2} \)
31 \( 1 + 3.34T + 31T^{2} \)
37 \( 1 + 5.32iT - 37T^{2} \)
41 \( 1 - 4.38T + 41T^{2} \)
43 \( 1 - 12.2iT - 43T^{2} \)
47 \( 1 - 9.54T + 47T^{2} \)
53 \( 1 + 10.6iT - 53T^{2} \)
59 \( 1 - 10.2iT - 59T^{2} \)
61 \( 1 + 13.2iT - 61T^{2} \)
67 \( 1 + 3.93iT - 67T^{2} \)
71 \( 1 - 6.95T + 71T^{2} \)
73 \( 1 + 5.93T + 73T^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 + 6.52iT - 83T^{2} \)
89 \( 1 - 6.69T + 89T^{2} \)
97 \( 1 - 3.99T + 97T^{2} \)
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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

−9.713221490170034665272327132948, −9.282034029983826003205618189377, −8.502921191288745479663416344050, −7.69556145356589666874966450644, −6.23789004048709129090366807212, −5.14417773083131138162404264736, −4.21893680979658266133672881122, −3.44031351647366409487204547096, −2.55132238845248669318751086039, −0.44203805577804098941225018369, 1.38461769662668654586874447687, 2.14636449955759814691807966042, 4.09062616720741030860980087805, 5.35172069930406926219673254331, 6.21604334041925689043479902887, 7.00281388768813495319184234983, 7.46856272630132693554146740940, 8.222867691749538731442877689510, 9.100074632878273339320733231916, 9.988506618297499728636860021126

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