↑

Properties

Label 2-800-100.23-c1-0-17
Degree $2$
Conductor $800$
Sign $0.980 + 0.195i$
Analytic cond. $6.38803$
Root an. cond. $2.52745$
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  + (0.802 + 1.57i)3-s + (1.58 − 1.57i)5-s + (−1.63 − 1.63i)7-s + (−0.0750 + 0.103i)9-s + (1.12 + 1.54i)11-s + (−0.802 − 5.06i)13-s + (3.75 + 1.23i)15-s + (5.98 + 3.04i)17-s + (−1.06 − 3.29i)19-s + (1.26 − 3.89i)21-s + (−0.167 + 1.05i)23-s + (0.0435 − 4.99i)25-s + (5.01 + 0.794i)27-s + (1.58 + 0.515i)29-s + (2.42 − 0.786i)31-s + ⋯
L(s)  = 1  + (0.463 + 0.909i)3-s + (0.710 − 0.704i)5-s + (−0.618 − 0.618i)7-s + (−0.0250 + 0.0344i)9-s + (0.338 + 0.466i)11-s + (−0.222 − 1.40i)13-s + (0.969 + 0.319i)15-s + (1.45 + 0.739i)17-s + (−0.245 − 0.755i)19-s + (0.276 − 0.849i)21-s + (−0.0349 + 0.220i)23-s + (0.00870 − 0.999i)25-s + (0.965 + 0.152i)27-s + (0.294 + 0.0957i)29-s + (0.434 − 0.141i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.980 + 0.195i$
Analytic conductor: \(6.38803\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ 0.980 + 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98860 - 0.196711i\)
\(L(\frac12)\) \(\approx\) \(1.98860 - 0.196711i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.58 + 1.57i)T \)
good3 \( 1 + (-0.802 - 1.57i)T + (-1.76 + 2.42i)T^{2} \)
7 \( 1 + (1.63 + 1.63i)T + 7iT^{2} \)
11 \( 1 + (-1.12 - 1.54i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.802 + 5.06i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-5.98 - 3.04i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (1.06 + 3.29i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.167 - 1.05i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (-1.58 - 0.515i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.42 + 0.786i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.41 + 0.382i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (3.16 + 2.29i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (5.65 - 5.65i)T - 43iT^{2} \)
47 \( 1 + (-2.00 + 1.02i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (10.8 - 5.54i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-9.07 - 6.58i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.66 - 1.21i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (4.46 - 8.77i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (-14.0 - 4.56i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.53 - 0.559i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-1.35 + 4.16i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (15.5 + 7.91i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (0.720 + 0.991i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-6.96 - 13.6i)T + (-57.0 + 78.4i)T^{2} \)
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

−10.00834480253722587459201599732, −9.677307420273849058303616679126, −8.702966758384767596372984383407, −7.86824843007547361457910216573, −6.70412560574575306500926130173, −5.67756643976375567313904016798, −4.76775602669232316040091047682, −3.79751071503121739830487608212, −2.87117261491998540542845778358, −1.07898676217973848318112682607, 1.57131447602521518966031271855, 2.54649462805372840950806670713, 3.49015183691575555635511326234, 5.10514149845473276244924597506, 6.27528353217181694693104811791, 6.68714249022214864459631722690, 7.64589477205833183447370757956, 8.556263034221086027820267137943, 9.550394886621654726054307932408, 10.01718941530591080541153378981

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