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Properties

Label 2-10e3-8.5-c1-0-76
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

\(L(1)\) \(\approx\) \(1.32017 - 1.04305i\)
\(L(\frac12)\) \(\approx\) \(1.32017 - 1.04305i\)
\(L(\frac{3}{2})\) 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

−9.369400983517919371104540787776, −8.585492608611252760075614959693, −7.941225092138832239858543150310, −7.11734998824596004347440909380, −6.49033141084776765299221199388, −5.78762327276722052495132163741, −4.76104333229179281303508417111, −3.39413556284903800138217184918, −2.37567826034005873128150281237, −0.64345409478402102281365805458, 1.74260888665745045218454706510, 3.17587972617589964912285158548, 3.87150624633248470921856695789, 4.84782682145323599537907289309, 5.26456949406516868786234283562, 6.45254914726948359101002981762, 7.79670178527630837427768062601, 8.980226329976036990864907859285, 9.690406512160410202213931757330, 10.13083796092785803877190238113

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