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Properties

Label 2-10e3-200.69-c1-0-36
Degree $2$
Conductor $1000$
Sign $0.995 + 0.0994i$
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  + (0.908 − 1.08i)2-s + (−2.01 + 1.46i)3-s + (−0.350 − 1.96i)4-s + (−0.243 + 3.51i)6-s + 0.110i·7-s + (−2.45 − 1.40i)8-s + (0.995 − 3.06i)9-s + (1.63 − 0.531i)11-s + (3.59 + 3.45i)12-s + (−1.48 + 4.56i)13-s + (0.120 + 0.100i)14-s + (−3.75 + 1.37i)16-s + (−0.269 + 0.371i)17-s + (−2.41 − 3.86i)18-s + (2.40 − 3.30i)19-s + ⋯
L(s)  = 1  + (0.642 − 0.766i)2-s + (−1.16 + 0.846i)3-s + (−0.175 − 0.984i)4-s + (−0.0993 + 1.43i)6-s + 0.0419i·7-s + (−0.867 − 0.497i)8-s + (0.331 − 1.02i)9-s + (0.493 − 0.160i)11-s + (1.03 + 0.998i)12-s + (−0.411 + 1.26i)13-s + (0.0321 + 0.0269i)14-s + (−0.938 + 0.344i)16-s + (−0.0653 + 0.0899i)17-s + (−0.569 − 0.910i)18-s + (0.550 − 0.757i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1000\)    =    \(2^{3} \cdot 5^{3}\)
Sign: $0.995 + 0.0994i$
Analytic conductor: \(7.98504\)
Root analytic conductor: \(2.82578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1000} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1000,\ (\ :1/2),\ 0.995 + 0.0994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39595 - 0.0695693i\)
\(L(\frac12)\) \(\approx\) \(1.39595 - 0.0695693i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.908 + 1.08i)T \)
5 \( 1 \)
good3 \( 1 + (2.01 - 1.46i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 - 0.110iT - 7T^{2} \)
11 \( 1 + (-1.63 + 0.531i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.48 - 4.56i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.269 - 0.371i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.40 + 3.30i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (-6.10 + 1.98i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-5.29 - 7.28i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.87 - 2.08i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.500 + 1.53i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.51 - 10.8i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 + (1.82 + 2.51i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.71 + 2.69i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.38 + 1.10i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-11.1 + 3.63i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (-3.02 - 2.19i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-9.04 + 6.57i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (9.85 - 3.20i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (4.45 - 3.23i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-9.91 - 7.20i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (2.07 + 6.38i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-3.30 - 4.54i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.20000491744595582035017112104, −9.421284369337402840256464541640, −8.787558460025455930332355393341, −6.91224796229391378458460230974, −6.40319446726048128993635193342, −5.21698074494298570567195490817, −4.77529254151383805083195189697, −3.92703840576068314355472198648, −2.69482864895067102272478333116, −1.04027102667241552223044638685, 0.798350797277698116365581233860, 2.66739829088019075666014445062, 3.98090056657524543749735337642, 5.16870866065491930315690663010, 5.70698871245784558982098778554, 6.49281598453529844082524312594, 7.31270524712118256855278641772, 7.84584985541771130595942930905, 8.950796748373182497376491897530, 10.09226664298466631733321406720

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