↑

Properties

Label 2-10e3-25.19-c1-0-4
Degree $2$
Conductor $1000$
Sign $0.999 - 0.0361i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.919 − 1.26i)3-s − 0.0338i·7-s + (0.170 − 0.523i)9-s + (1.79 + 5.51i)11-s + (2.96 + 0.964i)13-s + (−2.40 + 3.30i)17-s + (−0.00789 − 0.00573i)19-s + (−0.0429 + 0.0311i)21-s + (−2.20 + 0.717i)23-s + (−5.28 + 1.71i)27-s + (4.38 − 3.18i)29-s + (3.80 + 2.76i)31-s + (5.33 − 7.33i)33-s + (10.1 + 3.29i)37-s + (−1.50 − 4.64i)39-s + ⋯
L(s)  = 1  + (−0.531 − 0.731i)3-s − 0.0128i·7-s + (0.0566 − 0.174i)9-s + (0.540 + 1.66i)11-s + (0.822 + 0.267i)13-s + (−0.583 + 0.802i)17-s + (−0.00181 − 0.00131i)19-s + (−0.00936 + 0.00680i)21-s + (−0.460 + 0.149i)23-s + (−1.01 + 0.330i)27-s + (0.814 − 0.592i)29-s + (0.682 + 0.495i)31-s + (0.928 − 1.27i)33-s + (1.66 + 0.541i)37-s + (−0.241 − 0.743i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1000\)    =    \(2^{3} \cdot 5^{3}\)
Sign: $0.999 - 0.0361i$
Analytic conductor: \(7.98504\)
Root analytic conductor: \(2.82578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1000} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1000,\ (\ :1/2),\ 0.999 - 0.0361i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36110 + 0.0246241i\)
\(L(\frac12)\) \(\approx\) \(1.36110 + 0.0246241i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.919 + 1.26i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 + 0.0338iT - 7T^{2} \)
11 \( 1 + (-1.79 - 5.51i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-2.96 - 0.964i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.40 - 3.30i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.00789 + 0.00573i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (2.20 - 0.717i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-4.38 + 3.18i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-3.80 - 2.76i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-10.1 - 3.29i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.81 + 5.59i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 0.480iT - 43T^{2} \)
47 \( 1 + (-6.66 - 9.17i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (4.10 + 5.64i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.19 + 6.74i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.64 + 5.05i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-1.71 + 2.36i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (12.6 - 9.22i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-6.82 + 2.21i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (0.289 - 0.210i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.89 + 2.60i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (-4.71 - 14.4i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-1.26 - 1.73i)T + (-29.9 + 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.928266303432955045318269467045, −9.239745651337504503510905409547, −8.200874401688618982482102254632, −7.30804322959872063485659827196, −6.50435133557321436677401952622, −6.04360866129777074212599477042, −4.62087345130988271234856478964, −3.89672514993249295812856842657, −2.22110024394467391962757095456, −1.17196216468015958415060738413, 0.827359178766010126476775846800, 2.70049306824499353206503091239, 3.84521141686350077366486163391, 4.65449441578218338984636778377, 5.78714416618962721228638407258, 6.21980649651919555737487062120, 7.50723949178697147582692490913, 8.499927799963496989712965944462, 9.103630164345836286948954086177, 10.12955056745516294908273055092

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