↑

Properties

Label 2-20e2-80.67-c1-0-1
Degree $2$
Conductor $400$
Sign $-0.753 + 0.656i$
Analytic cond. $3.19401$
Root an. cond. $1.78718$
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  + (0.307 + 1.38i)2-s + 2.85i·3-s + (−1.81 + 0.849i)4-s + (−3.94 + 0.879i)6-s + (0.458 + 0.458i)7-s + (−1.73 − 2.23i)8-s − 5.15·9-s + (−0.492 − 0.492i)11-s + (−2.42 − 5.17i)12-s − 4.52·13-s + (−0.492 + 0.774i)14-s + (2.55 − 3.07i)16-s + (3.12 + 3.12i)17-s + (−1.58 − 7.11i)18-s + (4.04 + 4.04i)19-s + ⋯
L(s)  = 1  + (0.217 + 0.976i)2-s + 1.64i·3-s + (−0.905 + 0.424i)4-s + (−1.60 + 0.358i)6-s + (0.173 + 0.173i)7-s + (−0.611 − 0.791i)8-s − 1.71·9-s + (−0.148 − 0.148i)11-s + (−0.700 − 1.49i)12-s − 1.25·13-s + (−0.131 + 0.207i)14-s + (0.638 − 0.769i)16-s + (0.758 + 0.758i)17-s + (−0.374 − 1.67i)18-s + (0.928 + 0.928i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.753 + 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.753 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.753 + 0.656i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.753 + 0.656i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.391526 - 1.04526i\)
\(L(\frac12)\) \(\approx\) \(0.391526 - 1.04526i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.307 - 1.38i)T \)
5 \( 1 \)
good3 \( 1 - 2.85iT - 3T^{2} \)
7 \( 1 + (-0.458 - 0.458i)T + 7iT^{2} \)
11 \( 1 + (0.492 + 0.492i)T + 11iT^{2} \)
13 \( 1 + 4.52T + 13T^{2} \)
17 \( 1 + (-3.12 - 3.12i)T + 17iT^{2} \)
19 \( 1 + (-4.04 - 4.04i)T + 19iT^{2} \)
23 \( 1 + (-1.80 + 1.80i)T - 23iT^{2} \)
29 \( 1 + (3.83 - 3.83i)T - 29iT^{2} \)
31 \( 1 - 0.139iT - 31T^{2} \)
37 \( 1 + 5.84T + 37T^{2} \)
41 \( 1 - 4.55iT - 41T^{2} \)
43 \( 1 - 7.49T + 43T^{2} \)
47 \( 1 + (-4.14 + 4.14i)T - 47iT^{2} \)
53 \( 1 - 2.75iT - 53T^{2} \)
59 \( 1 + (3.62 - 3.62i)T - 59iT^{2} \)
61 \( 1 + (-3.72 - 3.72i)T + 61iT^{2} \)
67 \( 1 + 3.32T + 67T^{2} \)
71 \( 1 - 1.37T + 71T^{2} \)
73 \( 1 + (-2.55 - 2.55i)T + 73iT^{2} \)
79 \( 1 + 3.86T + 79T^{2} \)
83 \( 1 - 14.4iT - 83T^{2} \)
89 \( 1 + 3.35T + 89T^{2} \)
97 \( 1 + (-4.95 - 4.95i)T + 97iT^{2} \)
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

−11.87467324682018239539246936974, −10.56330574807322931562938350159, −9.886321504366818348258476557947, −9.146766399900070783759133535373, −8.222255414865048859094630148283, −7.22152672096311847438546057556, −5.68385265531010231587453092700, −5.18543164003541475276536575764, −4.13447517329387356535499221436, −3.18471313247461926987718762468, 0.69566009956929607529011409082, 2.08737956721368498379258887093, 3.07219917756937339987201163632, 4.84514896038918786083008130251, 5.78425971921459775534072139049, 7.24449592215304301625532196429, 7.66393798515085999133233184342, 9.008045239880111646636929034360, 9.832840806225848478694983837068, 11.08427173654377770327775885277

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