↑

Properties

Label 2-800-100.23-c1-0-12
Degree $2$
Conductor $800$
Sign $0.372 - 0.928i$
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  + (0.927 + 1.81i)3-s + (2.22 + 0.215i)5-s + (0.168 + 0.168i)7-s + (−0.687 + 0.946i)9-s + (−3.26 − 4.49i)11-s + (0.749 + 4.73i)13-s + (1.67 + 4.24i)15-s + (6.52 + 3.32i)17-s + (2.15 + 6.61i)19-s + (−0.150 + 0.461i)21-s + (0.984 − 6.21i)23-s + (4.90 + 0.958i)25-s + (3.69 + 0.584i)27-s + (−0.403 − 0.131i)29-s + (−6.52 + 2.12i)31-s + ⋯
L(s)  = 1  + (0.535 + 1.05i)3-s + (0.995 + 0.0963i)5-s + (0.0635 + 0.0635i)7-s + (−0.229 + 0.315i)9-s + (−0.985 − 1.35i)11-s + (0.207 + 1.31i)13-s + (0.431 + 1.09i)15-s + (1.58 + 0.805i)17-s + (0.493 + 1.51i)19-s + (−0.0327 + 0.100i)21-s + (0.205 − 1.29i)23-s + (0.981 + 0.191i)25-s + (0.710 + 0.112i)27-s + (−0.0749 − 0.0243i)29-s + (−1.17 + 0.380i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.372 - 0.928i$
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: \((2,\ 800,\ (\ :1/2),\ 0.372 - 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82390 + 1.23360i\)
\(L(\frac12)\) \(\approx\) \(1.82390 + 1.23360i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.22 - 0.215i)T \)
good3 \( 1 + (-0.927 - 1.81i)T + (-1.76 + 2.42i)T^{2} \)
7 \( 1 + (-0.168 - 0.168i)T + 7iT^{2} \)
11 \( 1 + (3.26 + 4.49i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.749 - 4.73i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-6.52 - 3.32i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-2.15 - 6.61i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.984 + 6.21i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (0.403 + 0.131i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.52 - 2.12i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (10.6 - 1.68i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (2.78 + 2.02i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (0.0127 - 0.0127i)T - 43iT^{2} \)
47 \( 1 + (3.27 - 1.66i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-3.62 + 1.84i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-5.38 - 3.90i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-7.47 + 5.43i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.62 + 3.19i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (5.30 + 1.72i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.26 - 0.517i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (0.104 - 0.321i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.12 + 1.08i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (5.36 + 7.38i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (8.01 + 15.7i)T + (-57.0 + 78.4i)T^{2} \)
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

−10.28101285086190195902469317735, −9.750322250054835595139307103413, −8.700478608685224002968420911631, −8.317702969134133816099383927567, −6.87097497278132420089174460884, −5.78344236447159658599032058693, −5.20627035201499055171963081376, −3.80111211160784193870110959612, −3.14560072311765084606731472117, −1.68788402078180863309109618581, 1.18519518660402666425249953773, 2.32716543449789946284126662584, 3.15718488095730443629650238795, 5.18558279747880878312420021235, 5.41938748732003879938202014539, 7.06470304350637994521662386780, 7.35733351880017274161037086526, 8.211461497034800024300553244184, 9.356277810979047560762386141868, 9.976254815602015448692975988712

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