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Properties

Label 2-10e3-25.9-c1-0-12
Degree $2$
Conductor $1000$
Sign $0.545 + 0.837i$
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  + (−2.12 + 0.690i)3-s − 0.381i·7-s + (1.61 − 1.17i)9-s + (2.61 + 1.90i)11-s + (−3.30 − 4.54i)13-s + (−0.726 − 0.236i)17-s + (−1.66 + 5.11i)19-s + (0.263 + 0.812i)21-s + (−2.04 + 2.80i)23-s + (1.31 − 1.80i)27-s + (−2.04 − 6.29i)29-s + (2.69 − 8.28i)31-s + (−6.88 − 2.23i)33-s + (−2.21 − 3.04i)37-s + (10.1 + 7.38i)39-s + ⋯
L(s)  = 1  + (−1.22 + 0.398i)3-s − 0.144i·7-s + (0.539 − 0.391i)9-s + (0.789 + 0.573i)11-s + (−0.915 − 1.26i)13-s + (−0.176 − 0.0572i)17-s + (−0.381 + 1.17i)19-s + (0.0575 + 0.177i)21-s + (−0.425 + 0.585i)23-s + (0.252 − 0.348i)27-s + (−0.379 − 1.16i)29-s + (0.483 − 1.48i)31-s + (−1.19 − 0.389i)33-s + (−0.363 − 0.500i)37-s + (1.62 + 1.18i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1000\)    =    \(2^{3} \cdot 5^{3}\)
Sign: $0.545 + 0.837i$
Analytic conductor: \(7.98504\)
Root analytic conductor: \(2.82578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1000} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1000,\ (\ :1/2),\ 0.545 + 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.639515 - 0.346632i\)
\(L(\frac12)\) \(\approx\) \(0.639515 - 0.346632i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (2.12 - 0.690i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + 0.381iT - 7T^{2} \)
11 \( 1 + (-2.61 - 1.90i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (3.30 + 4.54i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.726 + 0.236i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.66 - 5.11i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (2.04 - 2.80i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.04 + 6.29i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.69 + 8.28i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.21 + 3.04i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-6.23 + 4.53i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 6.61iT - 43T^{2} \)
47 \( 1 + (-6.51 + 2.11i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-9.45 + 3.07i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.73 + 2.71i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (7.42 + 5.39i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (0.277 + 0.0901i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (1.97 + 6.06i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-6.46 + 8.89i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.19 + 9.82i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (12.8 + 4.16i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-3.23 - 2.35i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-17.3 + 5.64i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.06741657250555374985629640392, −9.308293622269876647052187352336, −7.982472788179584646636967526694, −7.33479201922933572885662438853, −6.07678531383222372798575111001, −5.70212197468510398858321335240, −4.60505250596117353156518954943, −3.84373541173433722661425710627, −2.22837451168394019442492802713, −0.46070623345008273688206271035, 1.10716251054036764332486946467, 2.56043610465283469522534597633, 4.11882927446005178447232286845, 4.98919838073124951336664427520, 5.89006092815364380521883893361, 6.85016564393535544762916957148, 7.05131849095173322790427653319, 8.702229288408321992677864227004, 9.105435757521468962785876287399, 10.35526092414398122746334599348

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