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Properties

Label 2-10e3-1000.109-c1-0-108
Degree $2$
Conductor $1000$
Sign $-0.823 + 0.567i$
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  + (−1.21 − 0.716i)2-s + (−0.290 − 0.351i)3-s + (0.973 + 1.74i)4-s + (−1.00 − 1.99i)5-s + (0.102 + 0.637i)6-s + (3.27 − 1.06i)7-s + (0.0644 − 2.82i)8-s + (0.523 − 2.74i)9-s + (−0.197 + 3.15i)10-s + (0.505 − 4.00i)11-s + (0.331 − 0.850i)12-s + (0.348 − 1.82i)13-s + (−4.75 − 1.04i)14-s + (−0.407 + 0.935i)15-s + (−2.10 + 3.40i)16-s + (−0.351 − 1.36i)17-s + ⋯
L(s)  = 1  + (−0.862 − 0.506i)2-s + (−0.167 − 0.203i)3-s + (0.486 + 0.873i)4-s + (−0.451 − 0.892i)5-s + (0.0419 + 0.260i)6-s + (1.23 − 0.402i)7-s + (0.0227 − 0.999i)8-s + (0.174 − 0.914i)9-s + (−0.0626 + 0.998i)10-s + (0.152 − 1.20i)11-s + (0.0955 − 0.245i)12-s + (0.0967 − 0.507i)13-s + (−1.27 − 0.280i)14-s + (−0.105 + 0.241i)15-s + (−0.526 + 0.850i)16-s + (−0.0851 − 0.331i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.823 + 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.823 + 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1000\)    =    \(2^{3} \cdot 5^{3}\)
Sign: $-0.823 + 0.567i$
Analytic conductor: \(7.98504\)
Root analytic conductor: \(2.82578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1000} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1000,\ (\ :1/2),\ -0.823 + 0.567i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.306763 - 0.985160i\)
\(L(\frac12)\) \(\approx\) \(0.306763 - 0.985160i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.21 + 0.716i)T \)
5 \( 1 + (1.00 + 1.99i)T \)
good3 \( 1 + (0.290 + 0.351i)T + (-0.562 + 2.94i)T^{2} \)
7 \( 1 + (-3.27 + 1.06i)T + (5.66 - 4.11i)T^{2} \)
11 \( 1 + (-0.505 + 4.00i)T + (-10.6 - 2.73i)T^{2} \)
13 \( 1 + (-0.348 + 1.82i)T + (-12.0 - 4.78i)T^{2} \)
17 \( 1 + (0.351 + 1.36i)T + (-14.8 + 8.18i)T^{2} \)
19 \( 1 + (-4.46 - 3.69i)T + (3.56 + 18.6i)T^{2} \)
23 \( 1 + (2.55 - 2.72i)T + (-1.44 - 22.9i)T^{2} \)
29 \( 1 + (-8.15 + 0.513i)T + (28.7 - 3.63i)T^{2} \)
31 \( 1 + (-5.40 + 1.38i)T + (27.1 - 14.9i)T^{2} \)
37 \( 1 + (4.18 + 2.30i)T + (19.8 + 31.2i)T^{2} \)
41 \( 1 + (2.71 - 2.55i)T + (2.57 - 40.9i)T^{2} \)
43 \( 1 + (-0.804 + 0.584i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (2.83 + 7.16i)T + (-34.2 + 32.1i)T^{2} \)
53 \( 1 + (-0.588 + 0.928i)T + (-22.5 - 47.9i)T^{2} \)
59 \( 1 + (6.95 - 3.27i)T + (37.6 - 45.4i)T^{2} \)
61 \( 1 + (4.68 - 4.98i)T + (-3.83 - 60.8i)T^{2} \)
67 \( 1 + (-0.294 + 4.68i)T + (-66.4 - 8.39i)T^{2} \)
71 \( 1 + (0.689 - 0.272i)T + (51.7 - 48.6i)T^{2} \)
73 \( 1 + (-10.0 - 4.71i)T + (46.5 + 56.2i)T^{2} \)
79 \( 1 + (-1.98 - 2.39i)T + (-14.8 + 77.6i)T^{2} \)
83 \( 1 + (-7.58 + 9.17i)T + (-15.5 - 81.5i)T^{2} \)
89 \( 1 + (5.18 - 11.0i)T + (-56.7 - 68.5i)T^{2} \)
97 \( 1 + (15.5 - 0.979i)T + (96.2 - 12.1i)T^{2} \)
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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

−9.604615656296028198625198653452, −8.679557510799139730684871466183, −8.128455377567894832298516654340, −7.54420608558381377785856831274, −6.35770816228671114802112038990, −5.24902891151708909434754238569, −4.07367366134504435223997300460, −3.23422636317248438754569319874, −1.42483973775983267781510488203, −0.72956768678517023299788003394, 1.66245494530508043536470038812, 2.61532282373834278973624646854, 4.55659716535484214852059180247, 4.99704374186112115273651027907, 6.35689192246722128378649018701, 7.09435564060208675953437383566, 7.87495408049034452335287600894, 8.406015319727692011829045050098, 9.540042139774665932514289052460, 10.32651517909362779925777332612

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